Research Article Normal Forms of Hopf Bifurcation for a ...

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Research Article Normal Forms of Hopf Bifurcation for a Reaction-Diffusion System Subject to Neumann Boundary Condition Cun-Hua Zhang and Xiang-Ping Yan Department of Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, China Correspondence should be addressed to Xiang-Ping Yan; [email protected] Received 14 May 2015; Accepted 12 July 2015 Academic Editor: Alain Miranville Copyright © 2015 C.-H. Zhang and X.-P. Yan. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A reaction-diffusion system coupled by two equations subject to homogeneous Neumann boundary condition on one-dimensional spatial domain (0, ℓ) with ℓ>0 is considered. According to the normal form method and the center manifold theorem for reaction-diffusion equations, the explicit formulas determining the properties of Hopf bifurcation of spatially homogeneous and nonhomogeneous periodic solutions of system near the constant steady state (0, 0) are obtained. 1. Introduction As an important dynamic bifurcation phenomenon in dynamical systems, Hopf bifurcation of periodic solutions has attracted great interest of many authors in the last several decades [1–8]. In general, the study of Hopf bifurcation includes the existence and the properties such as the direc- tion of bifurcation and the stability of bifurcating periodic solutions. In application, however, it is more difficult to determine the properties of Hopf bifurcation than to find the existence of a Hopf bifurcation. An approach applied to determine the properties of Hopf bifurcation is to derive the projected equation of original equations on the associated center manifold, that is, the so-called normal form. en one may explore the local dynamical behaviors of a higher dimensional or even infinitely dimensional dynamical system near a certain nonhyperbolic steady state according to the normal form obtained. e normal form of Hopf bifurcation in ordinary differential equations (ODEs) with or without delays has been established well [1, 3, 5] since in this case the equilibrium is always constant and there are also no effects of spatial diffusion. Under some certain conditions, the reaction-diffusion equations under the homogeneous Neumann boundary con- dition may have the constant steady state and thus one can study the Hopf bifurcation of system at this constant steady state. Compared with the ODEs, it is more difficult to derive the normal form of Hopf bifurcation for reaction-diffusion equations at the constant steady state. Although Hassard et al. [3] established the method computing the normal form of Hopf bifurcation in reaction-diffusion equations with the homogeneous Neumann boundary condition and also considered the Hopf bifurcation of spatially homogeneous periodic solutions in Brusselator system, using the same method, Jin et al. [9] and Ruan [10] as well as Yi et al. [11, 12] considered the Hopf bifurcation of spatially homogeneous periodic solutions for Gierer-Meinhardt system and CIMA reaction, respectively. ere are few results regarding Hopf bifurcation of spatially nonhomogeneous periodic solutions for spatially homogeneous reaction-diffusion equations [7]. Based on the reason mentioned above, in this paper we consider the normal form of Hopf bifurcation of reaction- diffusion equations at the constant steady state following the idea in [3]. In order to have a clearer structure, we are concerned with the following general reaction-diffusion system coupled by two equations defined on one-dimensional spatial domain (0, ℓ) with ℓ> 0 and subject to Neumann boundary conditions; that is, = 1 + 1 (, , V), ∈(0, ℓ) , > 0, V = 2 V + 2 (, , V), ∈(0, ℓ) , > 0, (0, ) = V (0, ) = (ℓ, ) = V (ℓ, ) = 0, > 0, Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2015, Article ID 657307, 12 pages http://dx.doi.org/10.1155/2015/657307

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Research ArticleNormal Forms of Hopf Bifurcation for a Reaction-DiffusionSystem Subject to Neumann Boundary Condition

Cun-Hua Zhang and Xiang-Ping Yan

Department of Mathematics Lanzhou Jiaotong University Lanzhou 730070 China

Correspondence should be addressed to Xiang-Ping Yan xpyan72163com

Received 14 May 2015 Accepted 12 July 2015

Academic Editor Alain Miranville

Copyright copy 2015 C-H Zhang and X-P Yan This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

A reaction-diffusion system coupled by two equations subject to homogeneous Neumann boundary condition on one-dimensionalspatial domain (0 ℓ120587) with ℓ gt 0 is considered According to the normal form method and the center manifold theorem forreaction-diffusion equations the explicit formulas determining the properties of Hopf bifurcation of spatially homogeneous andnonhomogeneous periodic solutions of system near the constant steady state (0 0) are obtained

1 Introduction

As an important dynamic bifurcation phenomenon indynamical systems Hopf bifurcation of periodic solutionshas attracted great interest of many authors in the last severaldecades [1ndash8] In general the study of Hopf bifurcationincludes the existence and the properties such as the direc-tion of bifurcation and the stability of bifurcating periodicsolutions In application however it is more difficult todetermine the properties of Hopf bifurcation than to findthe existence of a Hopf bifurcation An approach applied todetermine the properties of Hopf bifurcation is to derive theprojected equation of original equations on the associatedcenter manifold that is the so-called normal form Thenone may explore the local dynamical behaviors of a higherdimensional or even infinitely dimensional dynamical systemnear a certain nonhyperbolic steady state according to thenormal form obtained The normal form of Hopf bifurcationin ordinary differential equations (ODEs) with or withoutdelays has been established well [1 3 5] since in this case theequilibrium is always constant and there are also no effects ofspatial diffusion

Under some certain conditions the reaction-diffusionequations under the homogeneous Neumann boundary con-dition may have the constant steady state and thus one canstudy the Hopf bifurcation of system at this constant steadystate Compared with the ODEs it is more difficult to derive

the normal form of Hopf bifurcation for reaction-diffusionequations at the constant steady state Although Hassardet al [3] established the method computing the normalformofHopf bifurcation in reaction-diffusion equationswiththe homogeneous Neumann boundary condition and alsoconsidered the Hopf bifurcation of spatially homogeneousperiodic solutions in Brusselator system using the samemethod Jin et al [9] and Ruan [10] as well as Yi et al [11 12]considered the Hopf bifurcation of spatially homogeneousperiodic solutions for Gierer-Meinhardt system and CIMAreaction respectively There are few results regarding Hopfbifurcation of spatially nonhomogeneous periodic solutionsfor spatially homogeneous reaction-diffusion equations [7]

Based on the reason mentioned above in this paper weconsider the normal form of Hopf bifurcation of reaction-diffusion equations at the constant steady state followingthe idea in [3] In order to have a clearer structure weare concerned with the following general reaction-diffusionsystem coupled by two equations defined onone-dimensionalspatial domain (0 ℓ120587) with ℓ gt 0 and subject to Neumannboundary conditions that is

119906119905= 1198891119906119909119909 +1198911 (120582 119906 V) 119909 isin (0 ℓ120587) 119905 gt 0

V119905= 1198892V119909119909 +1198912 (120582 119906 V) 119909 isin (0 ℓ120587) 119905 gt 0

119906119909(0 119905) = V

119909(0 119905) = 119906

119909(ℓ120587 119905) = V

119909(ℓ120587 119905) = 0

119905 gt 0

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2015 Article ID 657307 12 pageshttpdxdoiorg1011552015657307

2 Journal of Applied Mathematics

119906 (119909 0) = 1199060 (119909)

V (119909 0) = V0 (119909)

119909 isin (0 ℓ120587)

(1)in which 1198891 1198892 gt 0 are the diffusion coefficients 120582 isin R isthe parameter and 1198911 1198912 R times R2 rarr R are 119862119903 (119903 ge 5)functions with 119891

119896(120582 0 0) = 0 (119896 = 1 2) for any 120582 isin R

Although Yi et al [7] described the algorithm determiningthe properties of Hopf bifurcation of spatially homogeneousand nonhomogeneous periodic solutions for (1) at (0 0) andalso considered the Hopf bifurcation of a diffusive predator-prey system with Holling type-II functional response andsubject to the homogeneous Neumann boundary conditionthey did not give the normal form of Hopf bifurcationof spatially homogeneous and nonhomogeneous periodicsolutions of the general reaction-diffusion system (1) at (0 0)

This paper is organized as follows In the next sectionfollowing the abstract method according to [3] we describethe algorithm determining the properties of Hopf bifurcationof spatially homogeneous and nonhomogeneous periodicsolutions for system (1) at the constant steady state (0 0) InSection 3 the explicit formulas determining the properties ofHopf bifurcation of spatially homogeneous periodic solutionsfor system (1) at (0 0) are obtained The explicit formulasdetermining the properties of Hopf bifurcation of spatiallynonhomogeneous periodic solutions for (1) at (0 0) are alsoderived in Section 4

2 Algorithm Determining the Properties ofHopf Bifurcation

In this section we will describe the explicit algorithm deter-mining the direction of Hopf bifurcation and the stability ofthe bifurcating periodic solutions of system (1) at (0 0)

Define the real-valued Sobolev space 119883 by

119883 = (119906 V) isin1198672(0 ℓ120587) times119867

2(0 ℓ120587) | 119906

119909= V119909= 0 119909

= 0 ℓ120587

(2)

In terms of119883 the complex-valued Sobolev space119883C is givenby

119883C = 119883oplus 119894119883 = 1199091 + 1198941199092 1199091 1199092 isin119883 (3)and the inner product ⟨sdot sdot⟩ on 119883C is defined by

⟨1198801 1198802⟩ = intℓ120587

0(11990611199062 + V1V2) 119889119909

for 1198801 = (1199061 V1) isin 119883C 1198802 = (1199062 V2) isin 119883C

(4)

Let 119860(120582) = 1198911119906(120582 0 0) 119861(120582) = 1198911V(120582 0 0) 119862(120582) =

1198912119906(120582 0 0) and 119863(120582) = 1198912V(120582 0 0) and define the linearoperator 119871(120582) with the domain 119863

119871(120582)= 119883C by

119871 (120582) = (1198891

1205972

1205971199092 + 119860 (120582) 119861 (120582)

119862 (120582) 11988921205972

1205971199092 + 119863 (120582)

) (5)

Assume that for some 1205820 isin R the following condition holds

(H) There exists a neighborhood 119874 of 1205820 such that for120582 isin 119874 119871(120582) has a pair of simple and continuouslydifferentiable eigenvalues120572(120582)plusmn119894120596(120582)with 120572(1205820) = 0120596(1205820) = 1205960 gt 0 and 1205721015840(1205820) = 0 In addition all othereigenvalues of 119871(120582) have nonzero real parts for 120582 isin 119874

Then from [3 7] we know that system (1) undergoes a Hopfbifurcation at (0 0) when 120582 crosses through 1205820

Define the second-order matrix sequence 119871119895(120582) by

119871119895(120582) = (

119860 (120582) minus1198891119895

2

ℓ2119861 (120582)

119862 (120582) 119863 (120582) minus1198891119895

2

ℓ2

) 119895 isin N0 (6)

Then the characteristic equation of 119871119895(120582) is

1205732minus120573119879119895(120582) +119863

119895(120582) = 0 119895 isin N0 (7)

where

119879119895(120582) = 119860 (120582) +119863 (120582) minus

(1198891 + 1198892) 1198952

ℓ2

119863119895(120582) =

119889111988921198954

ℓ4minus (1198891119863 (120582) + 1198892119860 (120582))

1198952

ℓ2

+119860 (120582)119863 (120582) minus 119861 (120582) 119862 (120582)

(8)

The eigenvalues of 119871(120582) can be determined by theeigenvalues of 119871

119895(120582) (119895 isin N0) and we have the following

conclusion

Lemma 1 If 120573(120582) isin C is an eigenvalue of the operator 119871(120582)then there exists some 119899 isin N0 such that 120573(120582) is the eigenvalueof 119871119899(120582) and vice versa

Proof It is well known that the eigenvalue problem

minus12059310158401015840

= 120583120593

119909 isin (0 ℓ120587)

1205931015840(0) = 120593

1015840(ℓ120587) = 0

(9)

has eigenvalues 1198952ℓ2 (119895 = 0 1 2 ) with eigenfunctionscos(119895ℓ)119909 Assume that 120573(120582) isin C is an eigenvalue ofthe operator 119871(120582) and the corresponding eigenfunction is(120601 120595) isin 119883C that is

119871 (120582) (120601

120595) = 120582(

120601

120595) (10)

Notice that (120601 120595) isin 119883C can be represented as

(120601

120595) =

infin

sum119895=0

(119886119895

119887119895

) cos119895

ℓ119909 (11)

Journal of Applied Mathematics 3

where 119886119895 119887119895isin C (119895 isin N0) Then (10) can be written into

infin

sum119895=0

119871119895(120582)(

119886119895

119887119895

) cos119895

ℓ119909 = 120573 (120582)

infin

sum119895=0

(119886119895

119887119895

) cos119895

ℓ119909 (12)

From the orthogonality of the function sequencecos(119895ℓ)119909infin

119895=0 one can get from (12) that for each 119895 isin N0

119871119895(120582)(

119886119895

119887119895

) = 120573 (120582)(119886119895

119887119895

) (13)

Since (120601 120595) isin 119883C is the eigenfunction of 119871(120582) correspondingto the eigenvalue 120573(120582) (120601 120595) = 0 and so there must be some119899 isin N0 such that 0 = (119886

119899 119887119899) isin C times C Therefore 120573(120582) is the

eigenvalue of the matrix 119871119899

If 120573(120582) is the eigenvalue of some matrix 119871119899 then there

exists a nonzero vector (119886119899 119887119899) isin C times C such that (13) holds

Let

(120601

120595) = (

119886119899

119887119899

) cos 119899

ℓ119909 (14)

Then (120601 120595) = 0 and

119871 (120582) (120601

120595) = 119871 (120582) (

119886119899

119887119899

) cos 119899

ℓ119909 = 119871

119899(119886119899

119887119899

) cos 119899

ℓ119909

= 120582(119886119899

119887119899

) cos 119899

ℓ119909 = 120573 (120582) (

120601

120595)

(15)

This demonstrates that 120573(120582) is an eigenvalue of 119871(120582) and thusthe proof is complete

Lemma 1 shows that under assumption (H) there isa unique 119899 isin N0 such that plusmn1198941205960 are purely imaginaryeigenvalues of 119871

119899(1205820) that is 119879119899(1205820) = 0 and 119863

119899(1205820) gt 0

Furthermore it is easy to see that 119879119895(1205820) = 0 for any 119895 = 119899

Therefore 119871119895(1205820) (119895 = 119899) has eigenvalues with zero real parts

if and only if119863119895(1205820) = 0 Assume that 120573(120582) = 120572(120582) + 119894120596(120582) is

the eigenvalue of 119871(120582) for 120582 sufficiently approaching 1205820Thenby the smoothness of 119891

119896(119896 = 1 2) we know that 120573(120582) is also

the eigenvalue of 119871119899(120582) namely 120573(120582) satisfies the following

equation

1205732minus120573119879119899(120582) +119863

119899(120582) = 0 (16)

Under the assumption (H) differentiating the above equationwith respect to 120582 at 1205820 yields

119889120572 (1205820)

119889120582=12[1198601015840(1205820) +119863

1015840(1205820)] (17)

Based on the above discussion condition (H) has thefollowing equivalent form

119879119899(1205820) = 0

119863119899(1205820) gt 0

1198601015840(1205820) +119863

1015840(1205820) = 0

for some 119899 isin N0 119863119895(1205820) = 0 for any 119895 isin N0

(18)

Then we know that 1205960 = radic119863119899(1205820) and 119861(1205820) 119862(1205820) cannot

be equal to zero simultaneously when the hypothesis (H) issatisfied Therefore the eigenvector of 119871

119899(1205820) corresponding

to the eigenvalue 1198941205960 can be chosen as

(119886119899

119887119899

) = (

1

1198941205960 minus 119860 (1205820) + 11988911198992ℓ2

119861 (1205820)

) (19)

and thus the eigenfunction of 119871(1205820) corresponding to the

eigenvalue 1198941205960 has the form

119902 = (119886119899

119887119899

) cos 119899

ℓ119909

= (

1

1198941205960 minus 119860 (1205820) + 11988911198992ℓ2

119861 (1205820)

) cos 119899

ℓ119909

(20)

Let the linear operator 119871lowast(1205820) with the domain 119863119871lowast(1205820)

=

119883C be defined by

119871lowast(1205820) = (

11988911205972

1205971199092 + 119860 (1205820) 119862 (1205820)

119861 (1205820) 11988921205972

1205971199092 + 119863 (1205820)

) (21)

Then 119871lowast(1205820) is the adjoint operator of the operator119871(1205820) suchthat ⟨119880 119871(1205820)119881⟩ = ⟨119871lowast(1205820)119880 119881⟩ with 119880119881 isin 119883C Similarto the choice of the eigenfunction 119902 of the operator 119871(1205820)corresponding to the eigenvalue 1198941205960 we can choose

119902lowast= (

119886lowast119899

119887lowast119899

) cos 119899

ℓ119909

= (

1205960 + 119894 (119860 (1205820) minus 11988911198992ℓ2)

21205960 intℓ120587

0 cos2 (119899ℓ) 119909 119889119909

minus119894119861 (1205820)

21205960 intℓ120587

0 cos2 (119899ℓ) 119909 119889119909

) cos 119899

ℓ119909

(22)

such that119871lowast(1205820) 119902

lowast= minus 1198941205960119902

⟨119902lowast 119902⟩ = 1

⟨119902lowast 119902⟩ = 0

(23)

Define 119883119862 and 119883119878 by 119883119862 = 119911119902 + 119911119902 | 119911 isin C and 119883119878 = 119880 isin

119883 | ⟨119902lowast 119880⟩ = 0 respectively Then119883 can be decomposed asthe direct sum of 119883119862 and 119883119878 that is 119883 = 119883119862 oplus 119883119878 Thus forany 119880 = (119906 V) isin 119883 there exists 119911 isin C and 119908 = (1199081 1199082) isin 119883

119904

such that119880 = 119911119902+ 119911 119902 +119908

or

119906 = 119911119886119899cos 119899

ℓ119909 + 119911 119886

119899cos 119899

ℓ119909 + 1199081

V = 119911119887119899cos 119899

ℓ119909 + 119911119887

119899cos 119899

ℓ119909 + 1199082

(24)

4 Journal of Applied Mathematics

Define 119865(120582 119880) by

119865 (120582 119880) = (1198911 (120582 119906 V) minus 119860 (120582) 119906 minus 119861 (120582) V

1198912 (120582 119906 V) minus 119862 (120582) 119906 minus 119863 (120582) V) (25)

Then system (1) can be rewritten into the following abstractform

119889119880

119889119905= 119871 (120582)119880+119865 (120582 119880) (26)

When 120582 = 1205820 system (26) is reduced to

119889119880

119889119905= 119871 (1205820) 119880+1198650 (119880) (27)

where 1198650(119880) = 119865(120582 119880)|120582=1205820

In terms of (23) and decomposi-tion (24) system (27) can be transformed into the followingsystem in (119911 119908) coordinates

119889119911

119889119905= 1198941205960119911 + ⟨119902

lowast 1198650⟩

119889119908

119889119905= 119871 (1205820) 119908+119867 (119911 119911 119908)

(28)

where

119867(119911 119911 119908) = 1198650 minus ⟨119902lowast 1198650⟩ 119902 minus ⟨119902

lowast 1198650⟩ 119902

1198650 = 1198650 (119911119902 + 119911 119902 +119908) (29)

For 119883 = (1199091 1199092) 119884 = (1199101 1199102) and 119885 = (1199111 1199112) isin

119883C define the symmetric multilinear forms 119876(119883 119884) and119862(119883 119884 119885) respectively by

119876 (119883 119884) = (

2sum119896119895=1

12059721198911 (1205820 1205851 1205852)

120597120585119896120597120585119895

10038161003816100381610038161003816100381610038161003816100381610038161205851=1205852=0119909119896119910119895

2sum119896119895=1

12059721198912 (1205820 1205851 1205852)

120597120585119896120597120585119895

10038161003816100381610038161003816100381610038161003816100381610038161205851=1205852=0119909119896119910119895

) (30)

119862 (119883 119884 119885)

= (

2sum119896119895119897=1

12059731198911 (1205820 1205851 1205852)

120597120585119896120597120585119895120597120585119897

10038161003816100381610038161003816100381610038161003816100381610038161205851=1205852=0119909119896119910119895119911119897

2sum119896119895119897=1

12059731198912 (1205820 1205851 1205852)

120597120585119896120597120585119895120597120585119897

10038161003816100381610038161003816100381610038161003816100381610038161205851=1205852=0119909119896119910119895119911119897

)(31)

Then for 119880 = (119906 V) isin 119883 we have

1198650 (119880) =12119876 (119880119880) +

16119862 (119880119880119880) +119874 (|119880|

4) (32)

For the simplicity of notations we will use 119876119883119884

and 119862119883119884119885

todenote 119876(119883 119884) and 119862(119883 119884 119885) respectively

Let

119867(119911 119911 119908) =119867202

1199112+11986711119911119911 +

119867022

1199112+119874 (|119911|

3)

+119874 (|119911| |119908|)

(33)

Then from (29) and (32) one can get

11986720 = 119876119902119902

minus⟨119902lowast 119876119902119902⟩ 119902 minus ⟨119902

lowast 119876119902119902⟩ 119902

11986711 = 119876119902119902

minus⟨119902lowast 119876119902119902⟩ 119902 minus ⟨119902

lowast 119876119902119902⟩ 119902

(34)

From the center manifold theorem in [3] we can rewrite119908 inthe form

119908 =119908202

1199112+11990811119911119911 +

119908022

1199112+119874 (|119911|

3) (35)

The second equation of (28) (33) and (35) yields

11990820 = [21198941205960119868 minus 119871 (1205820)]minus1

11986720

11990811 = minus [119871 (1205820)]minus1

11986711(36)

Substituting (35) into the first equation of (28) gives theequation of reaction-diffusion system (1) restricted on thecenter manifold at (1205820 0 0) as

119889119911

119889119905= 1198941205960119911 + sum

2le119896+119895le3

119892119896119895

119896119895119911119896119911119895+119874 (|119911|

4) (37)

where 11989220 = ⟨119902lowast 119876119902119902⟩ 11989211 = ⟨119902lowast 119876

119902119902⟩ 11989202 = ⟨119902lowast 119876

119902 119902⟩ and

11989221 = 2 ⟨119902lowast 11987611990811119902

⟩+⟨119902lowast 11987611990820119902

⟩+⟨119902lowast 119862119902119902119902

⟩ (38)

The dynamics of (28) can be determined by the dynamics of(37)

In addition it can be observed from [3] that when 120582

approaches sufficiently 1205820 the Poincare normal form of (26)has the form

= (120572 (120582) + 119894120596 (120582)) 119911 + 119911

119872

sum119895=1

119888119895(120582) (119911119911)

119895 (39)

where 119911 is a complex variable119872 ge 1 and 119888119895(120582) are complex-

valued coefficients with

1198881 (1205820) =119894

21205960(1198922011989211 minus 2 100381610038161003816100381611989211

10038161003816100381610038162minus13100381610038161003816100381611989202

10038161003816100381610038162)+

119892212

=119894

21205960(⟨119902lowast 119876119902119902⟩ ⟨119902lowast 119876119902119902⟩minus 2 10038161003816100381610038161003816

⟨119902lowast 119876119902119902⟩10038161003816100381610038161003816

2

minus1310038161003816100381610038161003816⟨119902lowast 119876119902 119902

⟩10038161003816100381610038161003816

2)+⟨119902

lowast 11987611990811119902

⟩+12⟨119902lowast 11987611990820119902

⟩

+12⟨119902lowast 119862119902119902119902

⟩

(40)

The direction of Hopf bifurcation and the stability of thebifurcating periodic solutions of (1) at (1205820 0 0) can bedetermined by the sign of Re 1198881(1205820) andwe have the followingconclusion

Theorem 2 Assume that condition (H) (or equivalently (18))holdsThen system (1) undergoes a supercritical (or subcritical)Hopf bifurcation at (0 0) when 120582 = 1205820 if

11205721015840 (1205820)

Re 1198881 (1205820) lt 0 (119903119890119904119901 gt 0) (41)

In addition if all other eigenvalues of 119871(1205820) have negative realparts then the bifurcating periodic solutions are stable (respunstable) when Re 1198881(1205820) lt 0 (119903119890119904119901 gt 0)

Journal of Applied Mathematics 5

3 Spatially Homogeneous Hopf Bifurcation

From the description in the previous section we know thatHopf bifurcation of (1) at (1205820 0 0) is spatially homogeneousif condition (18) holds when 119899 = 0 In the present section wecompute Re 1198881(1205820) in (40) in order to determine the directionof spatially homogeneous Hopf bifurcation and the stabilityof bifurcating periodic solutions of (1) at (1205820 0 0) followingthe algorithm described in this pervious section

Lemma 3 If condition (18) is satisfied when 119899 = 0 then11986720 =

11986711 = 0

Proof From (20) and (22) one can see

1198860 = 1

1198870 =1198941205960 minus 119860 (1205820)

119861 (1205820)

119886lowast

0 =1205960 + 119894119860 (1205820)

2ℓ1205871205960

119887lowast

0 = minus119894119861 (1205820)

2ℓ1205871205960

(42)

where

1205960 = radic119860 (1205820)119863 (1205820) minus 119861 (1205820) 119862 (1205820)

= radicminus1198602 (1205820) minus 119861 (1205820) 119862 (1205820)

(43)

Let all the partial derivatives of 119891119896(120582 119906 V) (119896 = 1 2) be

evaluated at (1205820 0 0) and let 1198881198960 1198891198960 and 119890

1198960 (119896 = 1 2) bedefined respectively by

1198881198960 = 119891

119896119906119906+ 2119891119896119906V1198870 +119891

119896VV11988720

1198891198960 = 119891

119896119906119906+119891119896119906V (1198870 + 1198870) +119891

119896VV10038161003816100381610038161198870

10038161003816100381610038162

1198901198960 = 119891

119896119906119906119906+119891119896119906119906V (21198870 + 1198870) +119891

119896119906VV (210038161003816100381610038161198870

10038161003816100381610038162+ 119887

20 )

+119891119896VVV

10038161003816100381610038161198870100381610038161003816100381621198870

(44)

Then from (30) and (31) we can get

119876119902119902

= (11988810

11988820)

119876119902119902

= (11988910

11988920)

119862119902119902119902

= (11989010

11989020)

(45)

Therefore

⟨119902lowast 119876119902119902⟩ = int

ℓ120587

0(119886lowast

0 11988810 + 119887lowast

0 11988820)

= ℓ120587 (119886lowast

0 11988810 + 119887lowast

0 11988820)

⟨119902lowast 119876119902119902⟩ = int

ℓ120587

0(119886lowast

0 11988810 + 119887lowast

0 11988820)

= ℓ120587 (119886lowast

0 11988810 + 119887lowast

0 11988820)

⟨119902lowast 119876119902119902⟩ = int

ℓ120587

0(119886lowast

011988910 + 119887lowast

011988920)

= ℓ120587 (119886lowast

011988910 + 119887lowast

011988920)

⟨119902lowast 119876119902119902⟩ = int

ℓ120587

0(119886lowast

0 11988910 + 119887lowast

0 11988920)

= ℓ120587 (119886lowast

0 11988910 + 119887lowast

0 11988920)

(46)

From (34) and (46) one can obtain

11986720

= (

11988810 minus ℓ120587 [(119886lowast0 + 119886lowast

0 ) 11988810 + (119887lowast0 + 119887lowast

0) 11988820]

11988820 minus ℓ120587 [(119886lowast

01198870 + 119886lowast0 1198870) 11988810 + (119887lowast

01198870 + 119887lowast0 1198870) 11988820]

)

11986711

= (

11988910 minus ℓ120587 [(119886lowast0 + 119886lowast

0 ) 11988910 + (119887lowast0 + 119887lowast

0) 11988920]

11988920 minus ℓ120587 [(119886lowast

01198870 + 119886lowast0 1198870) 11988910 + (119887lowast

01198870 + 119887lowast0 1198870) 11988920]

)

(47)

Notice from (42) that

119886lowast

0 + 119886lowast

0 = 119887lowast

01198870 + 119887lowast

0 1198870 =1ℓ120587

119887lowast

0 + 119887lowast

0 = 119886lowast

01198870 + 119886lowast

0 1198870 = 0(48)

The conclusion follows by substituting (48) into (47)

Lemma 3 and (36) imply 11990820 = 11990811 = 0 when 119899 = 0 in(18) and thus we have

11989221 = ⟨119902lowast 119862119902119902119902

⟩ = intℓ120587

0(119886lowast

0 11989010 + 119887lowast

0 11989020)

= ℓ120587 (119886lowast

0 11989010 + 119887lowast

0 11989020)

(49)

It follows from (40) that

2 Re 1198881 (1205820) = Re( 119894

1205960⟨119902lowast 119876119902119902⟩ ⟨119902lowast 119876119902119902⟩

+⟨119902lowast 119862119902119902119902

⟩)

6 Journal of Applied Mathematics

= Re[

[

119894ℓ120587 (119886lowast

0 11988810 + 119887lowast

0 11988820) ℓ120587 (119886lowast

011988910 + 119887lowast

011988920)

1205960

+ ℓ120587 (119886lowast

0 11989010 + 119887lowast

0 11989020)]

]

(50)

We represent 119860(1205820) 119861(1205820) and 119862(1205820) by 119860 119861 and119862 respectively for the simplicity of notations and underassumption (18) with 119899 = 0 substituting 1198870 in (42) into (44)yields that for 119896 = 1 2

Re 1198881198960 =

119891119896119906119906

1198612 minus 2119891119896119906V119860119861 + 119891

119896VV (21198602 + 119861119862)

1198612

Im 1198881198960 = 21205960

119891119896119906V119861 minus 119891

119896VV119860

1198612

1198891198960 =

119891119896119906119906

119860 minus 2119891119896119906V119861 minus 119891

119896VV119862

119861

Re 1198901198960 =

119891119896119906119906119906

1198612minus 3119891119896119906119906V119860119861 + 2119891

119896119906VV1198602+ 119891119896VVV119860119862

1198612

Im 1198901198960 = 1205960

119891119896119906119906V119861 minus 2119891

119896119906VV119860 minus 119891119896VVV119862

1198612

(51)

From (42) (50) and (51) one can derive

2 Re 1198881 (1205820) =C0 (1205820)

412059620119861

2 (52)

where

C0 (1205820) = [11989111199061199061198602+ (1198912119906119906 minus 21198911119906V) 119860119861minus1198911VV119860119862

minus 21198912119906V1198612minus1198912VV119861119862] (1198911119906119906119861+1198911VV119862+ 21198912119906V119861

minus 21198912VV119860)+ (1198911119906119906119860minus 21198911119906V119861minus1198911VV119862) [1198911119906119906119860119861

+ 21198911119906V119861119862minus1198911VV119860119862+11989121199061199061198612minus 21198912119906V119860119861

+1198912VV (21198602+119861119862)] minus 2 (119860

2+119861119862)

sdot [(1198911119906119906119906 +1198912119906119906V) 1198612minus 2 (1198911119906119906V +1198912119906VV) 119860119861

minus1198912VVV119861119862]

(53)

Thus we have the following result

Theorem4 Assume that condition (18) is satisfied when 119899 = 0andC0(1205820) is defined by (53) Then the spatially homogeneousHopf bifurcation of system (1) at (1205820 0 0) is supercritical (respsubcritical) if

C0 (1205820)

1198601015840 (1205820) + 1198631015840 (1205820)lt 0 (119903119890119904119901 gt 0) (54)

Moreover if each eigenvalue of 119871119895(1205820) has negative real parts

for all 119895 isin N then the above spatially homogeneous bifurcatingperiodic solutions are stable (resp unstable) when

C0 (1205820) lt 0 (119903119890119904119901 gt 0) (55)

4 Spatially NonhomogeneousHopf Bifurcation

Notice that the spatially nonhomogeneous periodic solu-tions of (1) at (1205820 0 0) from Hopf bifurcation are unstableAccordingly in this section we will calculate Re 1198881(1205820) in (40)in order to determine the direction of Hopf bifurcation ofspatially nonhomogeneous periodic solutions of system (1) at(1205820 0 0) To this end we always assume that 119899 isin N in (18)throughout this section and still represent 119860(1205820) 119861(1205820) and119862(1205820) by119860 119861 and119862 respectivelyThus 119902lowast defined in (22) hasthe form

119902lowast= (

119886lowast119899

119887lowast119899

) cos 119899

ℓ119909

= (

1205960 + 119894 (119860 minus 11988911198992ℓ2)

ℓ1205871205960

minus119894119861

ℓ1205871205960

) cos 119899

ℓ119909

(56)

where

1205960 = radic(119860 minus1198891119899

2

ℓ2)(119863 minus

11988921198992

ℓ2) minus 119861119862

= radicminus(119860 minus1198891119899

2

ℓ2)

2

minus 119861119862

(57)

Since when 119899 isin N

intℓ120587

0cos3 119899

ℓ119909 119889119909 = 0 (58)

one can obtain

⟨119902lowast 119876119902119902⟩ = ⟨119902

lowast 119876119902119902⟩ = ⟨119902

lowast 119876119902119902⟩ = ⟨119902

lowast 119876119902119902⟩

= 0(59)

Thus in order to calculate Re 1198881(1205820) it remains to compute

⟨119902lowast 11987611990811119902

⟩

⟨119902lowast 11987611990820119902

⟩

⟨119902lowast 119862119902119902119902

⟩

(60)

Journal of Applied Mathematics 7

Let all the second- and third-order partial derivatives of119891119896(120582 119906 V) (119896 = 1 2) with respect to 119906 and V be evaluated at

(1205820 0 0) and let

119888119896119899

= 119891119896119906119906

+ 2119891119896119906V119887119899 +119891

119896VV1198872119899

119889119896119899

= 119891119896119906119906

+119891119896119906V (119887119899 + 119887

119899) +119891119896VV

100381610038161003816100381611988711989910038161003816100381610038162

119890119896119899

= 119891119896119906119906119906

+119891119896119906119906V (2119887119899 + 119887

119899) +119891119896119906VV (2

100381610038161003816100381611988711989910038161003816100381610038162+ 119887

2119899)

+119891119896VVV

100381610038161003816100381611988711989910038161003816100381610038162119887119899

119896 = 1 2

(61)

Then from (30) and (31) one can observe

119876119902119902

= (1198881119899

1198882119899) cos2 119899

ℓ119909 =

12(1198881119899

1198882119899)(1+ cos 2119899

ℓ119909)

119876119902119902

= (1198891119899

1198892119899) cos2 119899

ℓ119909 =

12(1198891119899

1198892119899)(1+ cos 2119899

ℓ119909)

119862119902119902119902

= (1198901119899

1198902119899) cos3 119899

ℓ119909

(62)

In view of (34) (59) and (62) we have

11986720 = 119876119902119902

=12(1198881119899

1198882119899)(1+ cos 2119899

ℓ119909)

11986711 = 119876119902119902

=12(1198891119899

1198892119899)(1+ cos 2119899

ℓ119909)

(63)

Equalities (63) show that the calculation of [21198941205960 minus 119871(1205820)]minus1

and [119871(1205820)]minus1 will be restricted on the subspaces spanned by

eigenmodes 1 and cos(2119899ℓ)119909Let

1205721 =(1211988911198892 minus 31198892

1) 1198994

ℓ4minus3 (1198892 minus 1198891) 1198601198992

ℓ2minus 31205962

0

1205722 =6 (1198891 + 1198892) 119899

21205960ℓ2

1205723 =11988921119899

4

ℓ4+

(1198892 minus 1198891) 1198601198992

ℓ2minus 31205962

0

1205724 = minus2 (1198891 + 1198892) 119899

21205960ℓ2

(64)

Then under condition (18) one can derive

[21198941205960119868 minus 1198712119899 (1205820)]minus1

=1

1205721 + 1198941205722

sdot(

21198941205960 + 119860 +(31198892 minus 1198891) 119899

2

ℓ2119861

119862 21198941205960 minus 119860 +41198891119899

2

ℓ2

)

[21198941205960119868 minus 1198710 (1205820)]minus1

=1

1205723 + 1198941205723

sdot(21198941205960 + 119860 minus

(1198891 + 1198892) 1198992

ℓ2119861

119862 21198941205960 minus 119860

)

(65)

From (36) and (61) we have

11990820 = [21198941205960119868 minus 1198712119899 (1205820)]

minus1

2cos 2119899

ℓ119909

+[21198941205960119868 minus 1198710 (1205820)]

minus1

2(

1198881119899

1198882119899) =

12 (1205721 + 1198941205722)

sdot(

[21198941205960 + 119860 +(31198892 minus 1198891) 119899

2

ℓ2] 1198881119899 + 1198611198882119899

1198621198881119899 + (21198941205960 minus 119860 +41198891119899

2

ℓ2) 1198882119899

)

sdot cos 2119899ℓ

119909 +1

2 (1205723 + 1198941205724)

sdot([21198941205960 + 119860 minus

(1198891 + 1198892) 1198992

ℓ2] 1198881119899 + 1198611198882119899

1198621198881119899 + (21198941205960 minus 119860) 1198882119899

)

(66)

Similarly we can get

11990811 =121205725

(

[119860 +(31198892 minus 1198891) 119899

2

ℓ2]1198891119899 + 1198611198892119899

1198621198891119899 + (41198891119899

2

ℓ2minus 119860)1198892119899

)

sdot cos 2119899ℓ

119909 +121205726

sdot([119860 minus

(1198891 + 1198892) 1198992

ℓ2]1198891119899 + 1198611198892119899

1198621198891119899 minus 1198601198892119899

)

(67)

where

1205725 =(1211988911198892 minus 31198892

1) 1198994

ℓ4minus3 (1198892 minus 1198891) 119860119899

2

ℓ2+120596

20

1205726 =11988921119899

4

ℓ4+

(1198892 minus 1198891) 1198601198992

ℓ2+120596

20

(68)

8 Journal of Applied Mathematics

From (30) we have

11987611990820119902

= (1198911119906119906120585 + 1198911119906V120578 + 1198911VV120574

1198912119906119906120585 + 1198912119906V120578 + 1198912VV120574) cos 119899

ℓ119909 cos 2119899

ℓ119909

+(1198911119906119906120591 + 1198911119906V120594 + 1198911VV120577

1198912119906119906120591 + 1198912119906V120594 + 1198912VV120577) cos 119899

ℓ119909

11987611990811119902

= (1198911119906119906120585 + 1198911119906V120578 + 1198911VV120574

1198912119906119906120585 + 1198912119906V120578 + 1198912VV120574) cos 119899

ℓ119909 cos 2119899

ℓ119909

+(1198911119906119906120591 + 1198911119906V120594 + 1198911VV120577

1198912119906119906120591 + 1198912119906V120594 + 1198912VV120577) cos 119899

ℓ119909

(69)

with

120585 =(21198941205960 + 119860 + (31198892 minus 1198891) 119899

2ℓ

2) 1198881119899 + 1198611198882119899

2 (1205721 + 1198941205722)

120578 =(21198941205960 + 119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860 minus 1198941205960) + 119861119862

2119861 (1205721 + 1198941205722)

sdot 1198881119899 +(51198891119899

2ℓ2 minus 2119860 + 1198941205960)

2 (1205721 + 1198941205722)1198882119899

120574 =[1198621198881119899 + (21198941205960 minus 119860 + 41198891119899

2ℓ2) 1198882119899] (11988911198992ℓ2 minus 119860 minus 1198941205960)

2119861 (1205721 + 1198941205722)

120591 =(21198941205960 + 119860 minus (1198891 + 1198892) 119899

2ℓ2) 1198881119899 + 1198611198882119899

2 (1205723 + 1198941205724)

120594 =(21198941205960 + 119860 minus (1198891 + 1198892) 119899

2ℓ2) (11988911198992ℓ2 minus 119860 minus 1198941205960) + 119861119862

2119861 (1205723 + 1198941205724)

sdot 1198881119899 +(1198891119899

2ℓ2 minus 2119860 + 1198941205960)

2 (1205723 + 1198941205724)1198882119899

120577 =[1198621198881119899 + (21198941205960 minus 119860) 1198882119899] (1198891119899

2ℓ2 minus 119860 minus 1198941205960)

2119861 (1205723 + 1198941205724)

120585 =(119860 + (31198892 minus 1198891) 119899

2ℓ2) 1198891119899 + 1198611198892119899

21205725

120578 =(119860 + (31198892 minus 1198891) 119899

2ℓ2) (1198941205960 minus 119860 + 11988911198992ℓ2) + 119861119862

211986112057251198891119899

+(1198941205960 minus 2119860 + 51198891119899

2ℓ2)

212057251198892119899

120574 =(1198941205960 minus 119860 + 1198891119899

2ℓ2) [1198621198891119899 + (411988911198992ℓ2 minus 119860) 1198892119899]

21198611205725

120591 =(119860 minus (1198891 + 1198892) 119899

2ℓ2) 1198891119899 + 1198611198892119899

21205726

120594 =(119860 minus (1198891 + 1198892) 119899

2ℓ2) (1198941205960 minus 119860 + 11988911198992ℓ2) + 119861119862

211986112057261198891119899

+(1198941205960 minus 2119860 + 1198891119899

2ℓ2)

212057261198892119899

120577 =(1198941205960 minus 119860 + 1198891119899

2ℓ2) (1198621198891119899 minus 1198601198892119899)

21198611205726

(70)

Notice that for 119899 isin N

intℓ120587

0cos2 119899

ℓ119909 119889119909 =

ℓ120587

2

intℓ120587

0cos 2119899

ℓ119909cos2 119899

ℓ119909 119889119909 =

ℓ120587

4

(71)

It follows from (69) that

⟨119902lowast 11987611990820119902

⟩ =ℓ120587

4[119886lowast

119899(1198911119906119906120585 +1198911119906V120578 +1198911VV120574)

+ 119887lowast

119899(1198912119906119906120585 +1198912119906V120578 +1198912VV120574)]

+ℓ120587

2[119886lowast

119899(1198911119906119906120591 +1198911119906V120594+1198911VV120577)

+ 119887lowast

119899(1198912119906119906120591 +1198912119906V120594+1198912VV120577)]

⟨119902lowast 11987611990811119902

⟩ =ℓ120587

4[119886lowast

119899(1198911119906119906120585 +1198911119906V120578 +1198911VV120574)

+ 119887lowast

119899(1198912119906119906120585 +1198912119906V120578 +1198912VV120574)]

+ℓ120587

2[119886lowast

119899(1198911119906119906120591 +1198911119906V120594+1198911VV120577)

+ 119887lowast

119899(1198912119906119906120591 +1198912119906V120594+1198912VV120577)]

(72)

Substituting 119887119899= (1198941205960minus119860(1205820)+1198891119899

2ℓ2)119861(1205820) into (61) gives

Re 119888119896119899

=119891119896119906119906

1198612 minus 2119891119896119906V (119860 minus 1198891119899

2ℓ2) 119861 + 119891119896VV [2 (119860 minus 1198891119899

2ℓ2)2+ 119861119862]

1198612

Im 119888119896119899

=21205960 [119891119896119906V119861 minus 119891

119896VV (119860 minus 11988911198992ℓ2)]

1198612

Journal of Applied Mathematics 9

119889119896119899

=119891119896119906119906

(119860 minus 11988911198992ℓ2) minus 2119891

119896119906V119861 minus 119891119896VV119862

119861

Re 119890119896119899

=119891119896119906119906119906

1198612 minus 3119891119896119906119906V (119860 minus 1198891119899

2ℓ2) 119861 + 2119891119896119906VV (119860 minus 1198891119899

2ℓ2)2+ 119891119896VVV (119860 minus 1198891119899

2ℓ2) 119862

1198612

Im 119890119896119899

=1205960 [119891119896119906119906V119861 minus 2119891

119896119906VV (119860 minus 11988911198992ℓ2) minus 119891

119896VVV119862]

1198612

119896 = 1 2(73)

Then from (70) and (73) we have

Re 120585 =[(119860 + (31198892 minus 1198891) 119899

2ℓ2)Re 1198881119899 + 119861Re 1198882119899 minus 21205960 Im 1198881119899] 1205721 + [(119860 + (31198892 minus 1198891) 119899

2ℓ2) Im 1198881119899 + 119861 Im 1198882119899 + 21205960 Re 1198881119899] 1205722

2 (12057221 + 1205722

2)

Im 120585 =[(119860 + (31198892 minus 1198891) 119899

2ℓ2) Im 1198881119899 + 119861 Im 1198882119899 + 21205960 Re 1198881119899] 1205721 minus [(119860 + (31198892 minus 1198891) 1198992ℓ2)Re 1198881119899 + 119861Re 1198882119899 minus 21205960 Im 1198881119899] 1205722

2 (12057221 + 1205722

2)

Re 120578

=[(119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0]Re 1198881119899 minus 31205960 ((1198891 minus 1198892) 1198992ℓ2 minus 119860) Im 1198881119899 + 119861 (51198891119899

2ℓ2 minus 2119860)Re 1198882119899 minus 1198611205960 Im 1198882119899

2119861 (12057221 + 1205722

2)

sdot 1205721

+[(119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0] Im 1198881119899 + 31205960 ((1198891 minus 1198892) 1198992ℓ2 minus 119860)Re 1198881119899 + 119861 (51198891119899

2ℓ2 minus 2119860) Im 1198882119899 + 1198611205960 Re 11988821198992119861 (1205722

1 + 12057222)

sdot 1205722

Im 120578

=[(119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0] Im 1198881119899 + 31205960 ((1198891 minus 1198892) 1198992ℓ2 minus 119860)Re 1198881119899 + 119861 (51198891119899

2ℓ2 minus 2119860) Im 1198882119899 + 1198611205960 Re 11988821198992119861 (1205722

1 + 12057222)

sdot 1205721

minus[(119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0]Re 1198881119899 minus 31205960 ((1198891 minus 1198892) 1198992ℓ2 minus 119860) Im 1198881119899 + 119861 (51198891119899

2ℓ2 minus 2119860)Re 1198882119899 minus 1198611205960 Im 1198882119899

2119861 (12057221 + 1205722

2)

sdot 1205722

Re 120574 =(1198891119899

2ℓ2 minus 119860) [119862Re 1198881119899 minus (119860 minus 411988911198992ℓ2)Re 1198882119899 minus 21205960 Im 1198882119899] + 1205960 [119862 Im 1198881119899 minus (119860 minus 41198891119899

2ℓ2) Im 1198882119899 + 21205960 Re 1198882119899]2119861 (1205722

1 + 12057222)

1205721

+(1198891119899

2ℓ2 minus 119860) [119862 Im 1198881119899 minus (119860 minus 411988911198992ℓ2) Im 1198882119899 + 21205960 Re 1198882119899] minus 1205960 [119862Re 1198881119899 minus (119860 minus 41198891119899

2ℓ2)Re 1198882119899 minus 21205960 Im 1198882119899]

2119861 (12057221 + 1205722

2)1205722

Im 120574 =(1198891119899

2ℓ2 minus 119860) [119862 Im 1198881119899 minus (119860 minus 411988911198992ℓ2) Im 1198882119899 + 21205960 Re 1198882119899] minus 1205960 [119862Re 1198881119899 minus (119860 minus 41198891119899

2ℓ2)Re 1198882119899 minus 21205960 Im 1198882119899]

2119861 (12057221 + 1205722

2)1205721

10 Journal of Applied Mathematics

minus(1198891119899

2ℓ2 minus 119860) [119862Re 1198881119899 minus (119860 minus 411988911198992ℓ2)Re 1198882119899 minus 21205960 Im 1198882119899] + 1205960 [119862 Im 1198881119899 minus (119860 minus 41198891119899

2ℓ2) Im 1198882119899 + 21205960 Re 1198882119899]2119861 (1205722

1 + 12057222)

1205722

Re 120591 =[(119860 + (1198891 + 1198892) 119899

2ℓ

2)Re 1198881119899 + 119861Re 1198882119899 minus 21205960 Im 1198881119899] 1205723 + [(119860 + (1198891 + 1198892) 119899

2ℓ

2) Im 1198881119899 + 119861 Im 1198882119899 + 21205960 Re 1198881119899] 1205724

2 (12057223 + 1205722

4)

Im 120591 =[(119860 + (1198891 + 1198892) 119899

2ℓ2) Im 1198881119899 + 119861 Im 1198882119899 + 21205960 Re 1198881119899] 1205723 minus [(119860 + (1198891 + 1198892) 1198992ℓ2)Re 1198881119899 + 119861Re 1198882119899 minus 21205960 Im 1198881119899] 1205724

2 (12057223 + 1205722

4)

Re120594

=[(119860 minus (1198891 + 1198892) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0]Re 1198881119899 minus 1205960 ((31198891 + 1198892) 1198992ℓ2 minus 3119860) Im 1198881119899 + 119861 (1198891119899

2ℓ2 minus 2119860)Re 1198882119899 minus 1198611205960 Im 1198882119899

2119861 (12057223 + 1205722

4)

sdot 1205723

+[(119860 minus (1198891 + 1198892) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0] Im 1198881119899 + 1205960 ((31198891 + 1198892) 1198992ℓ2 minus 3119860)Re 1198881119899 + 119861 (1198891119899

2ℓ2 minus 2119860) Im 1198882119899 + 1198611205960 Re 11988821198992119861 (1205722

3 + 12057224)

sdot 1205724

Im120594

=[(119860 minus (1198891 + 1198892) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0] Im 1198881119899 + 1205960 ((31198891 + 1198892) 1198992ℓ2 minus 3119860)Re 1198881119899 + 119861 (1198891119899

2ℓ2 minus 2119860) Im 1198882119899 + 1198611205960 Re 11988821198992119861 (1205722

3 + 12057224)

sdot 1205723

minus[(119860 minus (1198891 + 1198892) 119899

2ℓ

2) (1198891119899

2ℓ

2minus 119860) + 119861119862 + 21205962

0]Re 1198881119899 minus 1205960 ((31198891 + 1198892) 1198992ℓ

2minus 3119860) Im 1198881119899 + 119861 (1198891119899

2ℓ

2minus 2119860)Re 1198882119899 minus 1198611205960 Im 1198882119899

2119861 (12057223 + 1205722

4)

sdot 1205724

Re 120577 =(1198891119899

2ℓ

2minus 119860) (119862Re 1198881119899 minus 119860Re 1198882119899 minus 21205960 Im 1198882119899) + 1205960 (119862 Im 1198881119899 minus 119860 Im 1198882119899 + 21205960 Re 1198882119899)

2119861 (12057223 + 1205722

4)1205723

+(1198891119899

2ℓ2 minus 119860) (119862 Im 1198881119899 minus 119860 Im 1198882119899 + 21205960 Re 1198882119899) minus 1205960 (119862Re 1198881119899 minus 119860Re 1198882119899 minus 21205960 Im 1198882119899)

2119861 (12057223 + 1205722

4)1205724

Im 120577 =(1198891119899

2ℓ2 minus 119860) (119862 Im 1198881119899 minus 119860 Im 1198882119899 + 21205960 Re 1198882119899) minus 1205960 (119862Re 1198881119899 minus 119860Re 1198882119899 minus 21205960 Im 1198882119899)

2119861 (12057223 + 1205722

4)1205723

minus(1198891119899

2ℓ2 minus 119860) (119862Re 1198881119899 minus 119860Re 1198882119899 minus 21205960 Im 1198882119899) + 1205960 (119862 Im 1198881119899 minus 119860 Im 1198882119899 + 21205960 Re 1198882119899)2119861 (1205722

3 + 12057224)

1205724

Re 120578 =[(119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862] 1198891119899 + 119861 (51198891119899

2ℓ2 minus 2119860)1198892119899

21198611205725

Re 120578 =1205960 [(119860 + (31198892 minus 1198891) 119899

2ℓ2) 1198891119899 + 1198611198892119899]

21205725

Re 120574 =(1198891119899

2ℓ2 minus 119860) [1198621198891119899 + (411988911198992ℓ2 minus 119860) 1198892119899]

21198611205725

Im 120574 =1205960 [1198621198891119899 + (41198891119899

2ℓ2 minus 119860) 1198892119899]

21198611205725

Re120594 =[(119860 minus (1198891 + 1198892) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862] 1198891119899 + 119861 (1198891119899

2ℓ2 minus 2119860)1198892119899

21198611205726

Im120594 =1205960 [(119860 minus (1198891 + 1198892) 119899

2ℓ2) 1198891119899 + 1198611198892119899]

21205726119861 (1205820)

Re 120577 =(1198621198891119899 minus 1198601198892119899) (1198891119899

2ℓ

2minus 119860)

21198611205726

Im 120577 =1205960 (1198621198891119899 minus 1198601198892119899)

21198611205726

(74)

Journal of Applied Mathematics 11

Since ℓ120587119886lowast

119899= 1 minus 119894(119860 minus 1198891119899

2ℓ2)1205960 and ℓ120587119887lowast

119899= minus1198941198611205960

one can get

Re ⟨119902lowast 11987611990820119902

⟩ =14[1198911119906119906 (Re 120585 + 2 Re 120591)

+1198911119906V (Re 120578 + 2 Re120594) +1198911VV (Re 120574 + 2 Re 120577)]

+(119860 minus 1198891119899

2ℓ2)

41205960[1198911119906119906 (Im 120585 + 2 Im 120591)

+1198911119906V (Im 120578 + 2 Im120594) +1198911VV (Im 120574 + 2 Im 120577)]

+119861

41205960[1198912119906119906 (Im 120585 + 2 Im 120591)

+1198912119906V (Im 120578 + 2 Im120594) +1198912VV (Im 120574 + 2 Im 120577)]

Re ⟨119902lowast 11987611990811119902

⟩ =14[1198911119906119906 (120585 + 2120591)

+1198911119906V (Re 120578 + 2 Re120594) +1198911VV (Re 120574 + 2 Re 120577)]

+(119860 minus 1198891119899

2ℓ2)

41205960[1198911119906119906 (120585 + 2120591)

+1198911119906V (Re 120578 + 2 Re120594) +1198911VV (Re 120574 + 2 Re 120577)]

+119861

41205960[1198912119906V (Im 120578 + 2 Im120594)

+1198912VV (Im 120574 + 2 Im 120577)]

(75)

In addition it follows from intℓ120587

0 cos4(119899ℓ)119909 119889119909 = 3ℓ1205878and (62) that

⟨119902lowast 119862119902119902119902

⟩ =3ℓ1205878

(119886lowast

1198991198901119899 + 119887

lowast

1198991198902119899) (76)

Therefore

Re ⟨119902lowast 119862119902119902119902

⟩ =381198612 1198911119906119906119906119861

2

minus 31198911119906119906V (119860minus1198891119899

2

ℓ2)119861+ 21198911119906VV (119860minus

11988911198992

ℓ2)

2

+1198911VVV (119860minus1198891119899

2

ℓ2)119862+(119860minus

11988911198992

ℓ2)

sdot [1198911119906119906V119861minus 21198911119906VV (119860minus1198891119899

2

ℓ2)minus1198911VVV119862]

+119861[1198912119906119906V119861minus 21198912119906VV (119860minus1198891119899

2

ℓ2)minus1198912VVV119862]

(77)

Now by (40) we have

Re 1198881 (1205820) = Re ⟨119902lowast 11987611990811119902

⟩+12Re ⟨119902lowast 119876

11990820119902⟩+

12

sdotRe ⟨119902lowast 119862119902119902119902

⟩

=18[1198911119906119906 (Re 120585 + 2 Re 120591 + 2120585 + 4120591)

+1198911119906V (Re 120578 + 2 Re120594+ 2 Re 120578 + 4 Re120594)

+1198911VV (Re 120574 + 2 Re 120577 + 2 Re 120574 + 4 Re 120577)]

+(119860 minus 1198891119899

2ℓ2)

81205960[1198911119906119906 (Im 120585 + 2 Im 120591 + 2120585 + 4120591)

+1198911119906V (Im 120578 + 2 Im120594+ 2 Re 120578 + 4 Re120594)

+1198911VV (Im 120574 + 2 Im 120577 + 2 Re 120574 + 4 Re 120577)]

+119861

81205960[1198912119906119906 (Im 120585 + 2 Im 120591)

+1198912119906V (Im 120578119868+ 2 Im120594+ 2 Im 120578 + 4 Im120594)

+1198912VV (Im 120574 + Im 120577 + 2 Im 120574 + 4 Im 120577)]

+3

161198612 11989111199061199061199061198612minus 31198911119906119906V (119860minus

11988911198992

ℓ2)119861

+ 21198911119906VV (119860minus1198891119899

2

ℓ2)

2

+1198911VVV (119860minus1198891119899

2

ℓ2)119862

+(119860minus1198891119899

2

ℓ2)

sdot [1198911119906119906V119861minus 21198911119906VV (119860minus1198891119899

2

ℓ2)minus1198911VVV119862]

+119861[1198912119906119906V119861minus 21198912119906VV (119860minus1198891119899

2

ℓ2)minus1198912VVV119862]

(78)

Thus we have the following result

Theorem 5 Assume that condition (18) holds for 119899 isin N Thenthe spatially nonhomogeneous Hopf bifurcation of system (1) at(1205820 0 0) is supercritical (resp subcritical) if

Re 1198881 (1205820)

1198601015840 (1205820) + 1198631015840 (1205820)lt 0 (119903119890119904119901 gt 0) (79)

here Re 1198881(1205820) is given by (78)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

12 Journal of Applied Mathematics

Acknowledgments

Cun-Hua Zhang was supported by the Natural ScienceFoundation of Gansu Province (1212RJZA065) Xiang-PingYan was supported by the National Natural Science Foun-dation of China (11261028) Gansu Province Natural ScienceFoundation (145RJZA216) and China Scholarship Council

References

[1] A Andronov E A Leontovich I I Gordon and A G MaierTheory of Bifurcations of Dynamical Systems on a Plane IsraelProgram for Scientific Translations Jerusalem Israel 1971

[2] J Hale Theory of Functional Differential Equations SpringerNew York NY USA 1977

[3] B D Hassard N D Kazarinoff and Y H Wan Theory andApplications of Hopf Bifurcation Cambridge University PressCambridge UK 1981

[4] H Kielhofer Bifurcation Theory An Introduction with Appli-cations to PDEs vol 156 of Applied Mathematical SciencesSpringer New York NY USA 2004

[5] YA KuznetsovElements of Applied BifurcationTheory AppliedMathematical Sciences Springer New York NY USA 1998

[6] J-H Wu Theory and Applications of Partial Functional Differ-ential Equations Springer New York NY USA 1996

[7] F Yi JWei and J Shi ldquoBifurcation and spatiotemporal patternsin a homogeneous diffusive predator-prey systemrdquo Journal ofDifferential Equations vol 246 no 5 pp 1944ndash1977 2009

[8] L-J Yang and X-W Zeng ldquoStability of singular Hopf bifurca-tionsrdquo Journal of Differential Equations vol 206 no 1 pp 30ndash54 2004

[9] J Y Jin J P Shi J J Wei and F Q Yi ldquoBifurcations of patternedsolutions in the diffusive Lengyel-Epstein system of CIMAchemical reactionsrdquo Rocky Mountain Journal of Mathematicsvol 43 no 5 pp 1637ndash1674 2013

[10] S G Ruan ldquoDiffusion-driven instability in the Gierer-Mein-hardtmodel ofmorphogenesisrdquoNatural ResourceModeling vol11 no 2 pp 131ndash142 1998

[11] F Yi J Wei and J Shi ldquoDiffusion-driven instability andbifurcation in the Lengyel-Epstein systemrdquo Nonlinear AnalysisReal World Applications vol 9 no 3 pp 1038ndash1051 2008

[12] F Q Yi J J Wei and J P Shi ldquoGlobal asymptotical behavior ofthe Lengyel-Epstein reaction-diffusion systemrdquoApplied Mathe-matics Letters vol 22 no 1 pp 52ndash55 2009

Submit your manuscripts athttpwwwhindawicom

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Differential EquationsInternational Journal of

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Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 2: Research Article Normal Forms of Hopf Bifurcation for a ...

2 Journal of Applied Mathematics

119906 (119909 0) = 1199060 (119909)

V (119909 0) = V0 (119909)

119909 isin (0 ℓ120587)

(1)in which 1198891 1198892 gt 0 are the diffusion coefficients 120582 isin R isthe parameter and 1198911 1198912 R times R2 rarr R are 119862119903 (119903 ge 5)functions with 119891

119896(120582 0 0) = 0 (119896 = 1 2) for any 120582 isin R

Although Yi et al [7] described the algorithm determiningthe properties of Hopf bifurcation of spatially homogeneousand nonhomogeneous periodic solutions for (1) at (0 0) andalso considered the Hopf bifurcation of a diffusive predator-prey system with Holling type-II functional response andsubject to the homogeneous Neumann boundary conditionthey did not give the normal form of Hopf bifurcationof spatially homogeneous and nonhomogeneous periodicsolutions of the general reaction-diffusion system (1) at (0 0)

This paper is organized as follows In the next sectionfollowing the abstract method according to [3] we describethe algorithm determining the properties of Hopf bifurcationof spatially homogeneous and nonhomogeneous periodicsolutions for system (1) at the constant steady state (0 0) InSection 3 the explicit formulas determining the properties ofHopf bifurcation of spatially homogeneous periodic solutionsfor system (1) at (0 0) are obtained The explicit formulasdetermining the properties of Hopf bifurcation of spatiallynonhomogeneous periodic solutions for (1) at (0 0) are alsoderived in Section 4

2 Algorithm Determining the Properties ofHopf Bifurcation

In this section we will describe the explicit algorithm deter-mining the direction of Hopf bifurcation and the stability ofthe bifurcating periodic solutions of system (1) at (0 0)

Define the real-valued Sobolev space 119883 by

119883 = (119906 V) isin1198672(0 ℓ120587) times119867

2(0 ℓ120587) | 119906

119909= V119909= 0 119909

= 0 ℓ120587

(2)

In terms of119883 the complex-valued Sobolev space119883C is givenby

119883C = 119883oplus 119894119883 = 1199091 + 1198941199092 1199091 1199092 isin119883 (3)and the inner product ⟨sdot sdot⟩ on 119883C is defined by

⟨1198801 1198802⟩ = intℓ120587

0(11990611199062 + V1V2) 119889119909

for 1198801 = (1199061 V1) isin 119883C 1198802 = (1199062 V2) isin 119883C

(4)

Let 119860(120582) = 1198911119906(120582 0 0) 119861(120582) = 1198911V(120582 0 0) 119862(120582) =

1198912119906(120582 0 0) and 119863(120582) = 1198912V(120582 0 0) and define the linearoperator 119871(120582) with the domain 119863

119871(120582)= 119883C by

119871 (120582) = (1198891

1205972

1205971199092 + 119860 (120582) 119861 (120582)

119862 (120582) 11988921205972

1205971199092 + 119863 (120582)

) (5)

Assume that for some 1205820 isin R the following condition holds

(H) There exists a neighborhood 119874 of 1205820 such that for120582 isin 119874 119871(120582) has a pair of simple and continuouslydifferentiable eigenvalues120572(120582)plusmn119894120596(120582)with 120572(1205820) = 0120596(1205820) = 1205960 gt 0 and 1205721015840(1205820) = 0 In addition all othereigenvalues of 119871(120582) have nonzero real parts for 120582 isin 119874

Then from [3 7] we know that system (1) undergoes a Hopfbifurcation at (0 0) when 120582 crosses through 1205820

Define the second-order matrix sequence 119871119895(120582) by

119871119895(120582) = (

119860 (120582) minus1198891119895

2

ℓ2119861 (120582)

119862 (120582) 119863 (120582) minus1198891119895

2

ℓ2

) 119895 isin N0 (6)

Then the characteristic equation of 119871119895(120582) is

1205732minus120573119879119895(120582) +119863

119895(120582) = 0 119895 isin N0 (7)

where

119879119895(120582) = 119860 (120582) +119863 (120582) minus

(1198891 + 1198892) 1198952

ℓ2

119863119895(120582) =

119889111988921198954

ℓ4minus (1198891119863 (120582) + 1198892119860 (120582))

1198952

ℓ2

+119860 (120582)119863 (120582) minus 119861 (120582) 119862 (120582)

(8)

The eigenvalues of 119871(120582) can be determined by theeigenvalues of 119871

119895(120582) (119895 isin N0) and we have the following

conclusion

Lemma 1 If 120573(120582) isin C is an eigenvalue of the operator 119871(120582)then there exists some 119899 isin N0 such that 120573(120582) is the eigenvalueof 119871119899(120582) and vice versa

Proof It is well known that the eigenvalue problem

minus12059310158401015840

= 120583120593

119909 isin (0 ℓ120587)

1205931015840(0) = 120593

1015840(ℓ120587) = 0

(9)

has eigenvalues 1198952ℓ2 (119895 = 0 1 2 ) with eigenfunctionscos(119895ℓ)119909 Assume that 120573(120582) isin C is an eigenvalue ofthe operator 119871(120582) and the corresponding eigenfunction is(120601 120595) isin 119883C that is

119871 (120582) (120601

120595) = 120582(

120601

120595) (10)

Notice that (120601 120595) isin 119883C can be represented as

(120601

120595) =

infin

sum119895=0

(119886119895

119887119895

) cos119895

ℓ119909 (11)

Journal of Applied Mathematics 3

where 119886119895 119887119895isin C (119895 isin N0) Then (10) can be written into

infin

sum119895=0

119871119895(120582)(

119886119895

119887119895

) cos119895

ℓ119909 = 120573 (120582)

infin

sum119895=0

(119886119895

119887119895

) cos119895

ℓ119909 (12)

From the orthogonality of the function sequencecos(119895ℓ)119909infin

119895=0 one can get from (12) that for each 119895 isin N0

119871119895(120582)(

119886119895

119887119895

) = 120573 (120582)(119886119895

119887119895

) (13)

Since (120601 120595) isin 119883C is the eigenfunction of 119871(120582) correspondingto the eigenvalue 120573(120582) (120601 120595) = 0 and so there must be some119899 isin N0 such that 0 = (119886

119899 119887119899) isin C times C Therefore 120573(120582) is the

eigenvalue of the matrix 119871119899

If 120573(120582) is the eigenvalue of some matrix 119871119899 then there

exists a nonzero vector (119886119899 119887119899) isin C times C such that (13) holds

Let

(120601

120595) = (

119886119899

119887119899

) cos 119899

ℓ119909 (14)

Then (120601 120595) = 0 and

119871 (120582) (120601

120595) = 119871 (120582) (

119886119899

119887119899

) cos 119899

ℓ119909 = 119871

119899(119886119899

119887119899

) cos 119899

ℓ119909

= 120582(119886119899

119887119899

) cos 119899

ℓ119909 = 120573 (120582) (

120601

120595)

(15)

This demonstrates that 120573(120582) is an eigenvalue of 119871(120582) and thusthe proof is complete

Lemma 1 shows that under assumption (H) there isa unique 119899 isin N0 such that plusmn1198941205960 are purely imaginaryeigenvalues of 119871

119899(1205820) that is 119879119899(1205820) = 0 and 119863

119899(1205820) gt 0

Furthermore it is easy to see that 119879119895(1205820) = 0 for any 119895 = 119899

Therefore 119871119895(1205820) (119895 = 119899) has eigenvalues with zero real parts

if and only if119863119895(1205820) = 0 Assume that 120573(120582) = 120572(120582) + 119894120596(120582) is

the eigenvalue of 119871(120582) for 120582 sufficiently approaching 1205820Thenby the smoothness of 119891

119896(119896 = 1 2) we know that 120573(120582) is also

the eigenvalue of 119871119899(120582) namely 120573(120582) satisfies the following

equation

1205732minus120573119879119899(120582) +119863

119899(120582) = 0 (16)

Under the assumption (H) differentiating the above equationwith respect to 120582 at 1205820 yields

119889120572 (1205820)

119889120582=12[1198601015840(1205820) +119863

1015840(1205820)] (17)

Based on the above discussion condition (H) has thefollowing equivalent form

119879119899(1205820) = 0

119863119899(1205820) gt 0

1198601015840(1205820) +119863

1015840(1205820) = 0

for some 119899 isin N0 119863119895(1205820) = 0 for any 119895 isin N0

(18)

Then we know that 1205960 = radic119863119899(1205820) and 119861(1205820) 119862(1205820) cannot

be equal to zero simultaneously when the hypothesis (H) issatisfied Therefore the eigenvector of 119871

119899(1205820) corresponding

to the eigenvalue 1198941205960 can be chosen as

(119886119899

119887119899

) = (

1

1198941205960 minus 119860 (1205820) + 11988911198992ℓ2

119861 (1205820)

) (19)

and thus the eigenfunction of 119871(1205820) corresponding to the

eigenvalue 1198941205960 has the form

119902 = (119886119899

119887119899

) cos 119899

ℓ119909

= (

1

1198941205960 minus 119860 (1205820) + 11988911198992ℓ2

119861 (1205820)

) cos 119899

ℓ119909

(20)

Let the linear operator 119871lowast(1205820) with the domain 119863119871lowast(1205820)

=

119883C be defined by

119871lowast(1205820) = (

11988911205972

1205971199092 + 119860 (1205820) 119862 (1205820)

119861 (1205820) 11988921205972

1205971199092 + 119863 (1205820)

) (21)

Then 119871lowast(1205820) is the adjoint operator of the operator119871(1205820) suchthat ⟨119880 119871(1205820)119881⟩ = ⟨119871lowast(1205820)119880 119881⟩ with 119880119881 isin 119883C Similarto the choice of the eigenfunction 119902 of the operator 119871(1205820)corresponding to the eigenvalue 1198941205960 we can choose

119902lowast= (

119886lowast119899

119887lowast119899

) cos 119899

ℓ119909

= (

1205960 + 119894 (119860 (1205820) minus 11988911198992ℓ2)

21205960 intℓ120587

0 cos2 (119899ℓ) 119909 119889119909

minus119894119861 (1205820)

21205960 intℓ120587

0 cos2 (119899ℓ) 119909 119889119909

) cos 119899

ℓ119909

(22)

such that119871lowast(1205820) 119902

lowast= minus 1198941205960119902

⟨119902lowast 119902⟩ = 1

⟨119902lowast 119902⟩ = 0

(23)

Define 119883119862 and 119883119878 by 119883119862 = 119911119902 + 119911119902 | 119911 isin C and 119883119878 = 119880 isin

119883 | ⟨119902lowast 119880⟩ = 0 respectively Then119883 can be decomposed asthe direct sum of 119883119862 and 119883119878 that is 119883 = 119883119862 oplus 119883119878 Thus forany 119880 = (119906 V) isin 119883 there exists 119911 isin C and 119908 = (1199081 1199082) isin 119883

119904

such that119880 = 119911119902+ 119911 119902 +119908

or

119906 = 119911119886119899cos 119899

ℓ119909 + 119911 119886

119899cos 119899

ℓ119909 + 1199081

V = 119911119887119899cos 119899

ℓ119909 + 119911119887

119899cos 119899

ℓ119909 + 1199082

(24)

4 Journal of Applied Mathematics

Define 119865(120582 119880) by

119865 (120582 119880) = (1198911 (120582 119906 V) minus 119860 (120582) 119906 minus 119861 (120582) V

1198912 (120582 119906 V) minus 119862 (120582) 119906 minus 119863 (120582) V) (25)

Then system (1) can be rewritten into the following abstractform

119889119880

119889119905= 119871 (120582)119880+119865 (120582 119880) (26)

When 120582 = 1205820 system (26) is reduced to

119889119880

119889119905= 119871 (1205820) 119880+1198650 (119880) (27)

where 1198650(119880) = 119865(120582 119880)|120582=1205820

In terms of (23) and decomposi-tion (24) system (27) can be transformed into the followingsystem in (119911 119908) coordinates

119889119911

119889119905= 1198941205960119911 + ⟨119902

lowast 1198650⟩

119889119908

119889119905= 119871 (1205820) 119908+119867 (119911 119911 119908)

(28)

where

119867(119911 119911 119908) = 1198650 minus ⟨119902lowast 1198650⟩ 119902 minus ⟨119902

lowast 1198650⟩ 119902

1198650 = 1198650 (119911119902 + 119911 119902 +119908) (29)

For 119883 = (1199091 1199092) 119884 = (1199101 1199102) and 119885 = (1199111 1199112) isin

119883C define the symmetric multilinear forms 119876(119883 119884) and119862(119883 119884 119885) respectively by

119876 (119883 119884) = (

2sum119896119895=1

12059721198911 (1205820 1205851 1205852)

120597120585119896120597120585119895

10038161003816100381610038161003816100381610038161003816100381610038161205851=1205852=0119909119896119910119895

2sum119896119895=1

12059721198912 (1205820 1205851 1205852)

120597120585119896120597120585119895

10038161003816100381610038161003816100381610038161003816100381610038161205851=1205852=0119909119896119910119895

) (30)

119862 (119883 119884 119885)

= (

2sum119896119895119897=1

12059731198911 (1205820 1205851 1205852)

120597120585119896120597120585119895120597120585119897

10038161003816100381610038161003816100381610038161003816100381610038161205851=1205852=0119909119896119910119895119911119897

2sum119896119895119897=1

12059731198912 (1205820 1205851 1205852)

120597120585119896120597120585119895120597120585119897

10038161003816100381610038161003816100381610038161003816100381610038161205851=1205852=0119909119896119910119895119911119897

)(31)

Then for 119880 = (119906 V) isin 119883 we have

1198650 (119880) =12119876 (119880119880) +

16119862 (119880119880119880) +119874 (|119880|

4) (32)

For the simplicity of notations we will use 119876119883119884

and 119862119883119884119885

todenote 119876(119883 119884) and 119862(119883 119884 119885) respectively

Let

119867(119911 119911 119908) =119867202

1199112+11986711119911119911 +

119867022

1199112+119874 (|119911|

3)

+119874 (|119911| |119908|)

(33)

Then from (29) and (32) one can get

11986720 = 119876119902119902

minus⟨119902lowast 119876119902119902⟩ 119902 minus ⟨119902

lowast 119876119902119902⟩ 119902

11986711 = 119876119902119902

minus⟨119902lowast 119876119902119902⟩ 119902 minus ⟨119902

lowast 119876119902119902⟩ 119902

(34)

From the center manifold theorem in [3] we can rewrite119908 inthe form

119908 =119908202

1199112+11990811119911119911 +

119908022

1199112+119874 (|119911|

3) (35)

The second equation of (28) (33) and (35) yields

11990820 = [21198941205960119868 minus 119871 (1205820)]minus1

11986720

11990811 = minus [119871 (1205820)]minus1

11986711(36)

Substituting (35) into the first equation of (28) gives theequation of reaction-diffusion system (1) restricted on thecenter manifold at (1205820 0 0) as

119889119911

119889119905= 1198941205960119911 + sum

2le119896+119895le3

119892119896119895

119896119895119911119896119911119895+119874 (|119911|

4) (37)

where 11989220 = ⟨119902lowast 119876119902119902⟩ 11989211 = ⟨119902lowast 119876

119902119902⟩ 11989202 = ⟨119902lowast 119876

119902 119902⟩ and

11989221 = 2 ⟨119902lowast 11987611990811119902

⟩+⟨119902lowast 11987611990820119902

⟩+⟨119902lowast 119862119902119902119902

⟩ (38)

The dynamics of (28) can be determined by the dynamics of(37)

In addition it can be observed from [3] that when 120582

approaches sufficiently 1205820 the Poincare normal form of (26)has the form

= (120572 (120582) + 119894120596 (120582)) 119911 + 119911

119872

sum119895=1

119888119895(120582) (119911119911)

119895 (39)

where 119911 is a complex variable119872 ge 1 and 119888119895(120582) are complex-

valued coefficients with

1198881 (1205820) =119894

21205960(1198922011989211 minus 2 100381610038161003816100381611989211

10038161003816100381610038162minus13100381610038161003816100381611989202

10038161003816100381610038162)+

119892212

=119894

21205960(⟨119902lowast 119876119902119902⟩ ⟨119902lowast 119876119902119902⟩minus 2 10038161003816100381610038161003816

⟨119902lowast 119876119902119902⟩10038161003816100381610038161003816

2

minus1310038161003816100381610038161003816⟨119902lowast 119876119902 119902

⟩10038161003816100381610038161003816

2)+⟨119902

lowast 11987611990811119902

⟩+12⟨119902lowast 11987611990820119902

⟩

+12⟨119902lowast 119862119902119902119902

⟩

(40)

The direction of Hopf bifurcation and the stability of thebifurcating periodic solutions of (1) at (1205820 0 0) can bedetermined by the sign of Re 1198881(1205820) andwe have the followingconclusion

Theorem 2 Assume that condition (H) (or equivalently (18))holdsThen system (1) undergoes a supercritical (or subcritical)Hopf bifurcation at (0 0) when 120582 = 1205820 if

11205721015840 (1205820)

Re 1198881 (1205820) lt 0 (119903119890119904119901 gt 0) (41)

In addition if all other eigenvalues of 119871(1205820) have negative realparts then the bifurcating periodic solutions are stable (respunstable) when Re 1198881(1205820) lt 0 (119903119890119904119901 gt 0)

Journal of Applied Mathematics 5

3 Spatially Homogeneous Hopf Bifurcation

From the description in the previous section we know thatHopf bifurcation of (1) at (1205820 0 0) is spatially homogeneousif condition (18) holds when 119899 = 0 In the present section wecompute Re 1198881(1205820) in (40) in order to determine the directionof spatially homogeneous Hopf bifurcation and the stabilityof bifurcating periodic solutions of (1) at (1205820 0 0) followingthe algorithm described in this pervious section

Lemma 3 If condition (18) is satisfied when 119899 = 0 then11986720 =

11986711 = 0

Proof From (20) and (22) one can see

1198860 = 1

1198870 =1198941205960 minus 119860 (1205820)

119861 (1205820)

119886lowast

0 =1205960 + 119894119860 (1205820)

2ℓ1205871205960

119887lowast

0 = minus119894119861 (1205820)

2ℓ1205871205960

(42)

where

1205960 = radic119860 (1205820)119863 (1205820) minus 119861 (1205820) 119862 (1205820)

= radicminus1198602 (1205820) minus 119861 (1205820) 119862 (1205820)

(43)

Let all the partial derivatives of 119891119896(120582 119906 V) (119896 = 1 2) be

evaluated at (1205820 0 0) and let 1198881198960 1198891198960 and 119890

1198960 (119896 = 1 2) bedefined respectively by

1198881198960 = 119891

119896119906119906+ 2119891119896119906V1198870 +119891

119896VV11988720

1198891198960 = 119891

119896119906119906+119891119896119906V (1198870 + 1198870) +119891

119896VV10038161003816100381610038161198870

10038161003816100381610038162

1198901198960 = 119891

119896119906119906119906+119891119896119906119906V (21198870 + 1198870) +119891

119896119906VV (210038161003816100381610038161198870

10038161003816100381610038162+ 119887

20 )

+119891119896VVV

10038161003816100381610038161198870100381610038161003816100381621198870

(44)

Then from (30) and (31) we can get

119876119902119902

= (11988810

11988820)

119876119902119902

= (11988910

11988920)

119862119902119902119902

= (11989010

11989020)

(45)

Therefore

⟨119902lowast 119876119902119902⟩ = int

ℓ120587

0(119886lowast

0 11988810 + 119887lowast

0 11988820)

= ℓ120587 (119886lowast

0 11988810 + 119887lowast

0 11988820)

⟨119902lowast 119876119902119902⟩ = int

ℓ120587

0(119886lowast

0 11988810 + 119887lowast

0 11988820)

= ℓ120587 (119886lowast

0 11988810 + 119887lowast

0 11988820)

⟨119902lowast 119876119902119902⟩ = int

ℓ120587

0(119886lowast

011988910 + 119887lowast

011988920)

= ℓ120587 (119886lowast

011988910 + 119887lowast

011988920)

⟨119902lowast 119876119902119902⟩ = int

ℓ120587

0(119886lowast

0 11988910 + 119887lowast

0 11988920)

= ℓ120587 (119886lowast

0 11988910 + 119887lowast

0 11988920)

(46)

From (34) and (46) one can obtain

11986720

= (

11988810 minus ℓ120587 [(119886lowast0 + 119886lowast

0 ) 11988810 + (119887lowast0 + 119887lowast

0) 11988820]

11988820 minus ℓ120587 [(119886lowast

01198870 + 119886lowast0 1198870) 11988810 + (119887lowast

01198870 + 119887lowast0 1198870) 11988820]

)

11986711

= (

11988910 minus ℓ120587 [(119886lowast0 + 119886lowast

0 ) 11988910 + (119887lowast0 + 119887lowast

0) 11988920]

11988920 minus ℓ120587 [(119886lowast

01198870 + 119886lowast0 1198870) 11988910 + (119887lowast

01198870 + 119887lowast0 1198870) 11988920]

)

(47)

Notice from (42) that

119886lowast

0 + 119886lowast

0 = 119887lowast

01198870 + 119887lowast

0 1198870 =1ℓ120587

119887lowast

0 + 119887lowast

0 = 119886lowast

01198870 + 119886lowast

0 1198870 = 0(48)

The conclusion follows by substituting (48) into (47)

Lemma 3 and (36) imply 11990820 = 11990811 = 0 when 119899 = 0 in(18) and thus we have

11989221 = ⟨119902lowast 119862119902119902119902

⟩ = intℓ120587

0(119886lowast

0 11989010 + 119887lowast

0 11989020)

= ℓ120587 (119886lowast

0 11989010 + 119887lowast

0 11989020)

(49)

It follows from (40) that

2 Re 1198881 (1205820) = Re( 119894

1205960⟨119902lowast 119876119902119902⟩ ⟨119902lowast 119876119902119902⟩

+⟨119902lowast 119862119902119902119902

⟩)

6 Journal of Applied Mathematics

= Re[

[

119894ℓ120587 (119886lowast

0 11988810 + 119887lowast

0 11988820) ℓ120587 (119886lowast

011988910 + 119887lowast

011988920)

1205960

+ ℓ120587 (119886lowast

0 11989010 + 119887lowast

0 11989020)]

]

(50)

We represent 119860(1205820) 119861(1205820) and 119862(1205820) by 119860 119861 and119862 respectively for the simplicity of notations and underassumption (18) with 119899 = 0 substituting 1198870 in (42) into (44)yields that for 119896 = 1 2

Re 1198881198960 =

119891119896119906119906

1198612 minus 2119891119896119906V119860119861 + 119891

119896VV (21198602 + 119861119862)

1198612

Im 1198881198960 = 21205960

119891119896119906V119861 minus 119891

119896VV119860

1198612

1198891198960 =

119891119896119906119906

119860 minus 2119891119896119906V119861 minus 119891

119896VV119862

119861

Re 1198901198960 =

119891119896119906119906119906

1198612minus 3119891119896119906119906V119860119861 + 2119891

119896119906VV1198602+ 119891119896VVV119860119862

1198612

Im 1198901198960 = 1205960

119891119896119906119906V119861 minus 2119891

119896119906VV119860 minus 119891119896VVV119862

1198612

(51)

From (42) (50) and (51) one can derive

2 Re 1198881 (1205820) =C0 (1205820)

412059620119861

2 (52)

where

C0 (1205820) = [11989111199061199061198602+ (1198912119906119906 minus 21198911119906V) 119860119861minus1198911VV119860119862

minus 21198912119906V1198612minus1198912VV119861119862] (1198911119906119906119861+1198911VV119862+ 21198912119906V119861

minus 21198912VV119860)+ (1198911119906119906119860minus 21198911119906V119861minus1198911VV119862) [1198911119906119906119860119861

+ 21198911119906V119861119862minus1198911VV119860119862+11989121199061199061198612minus 21198912119906V119860119861

+1198912VV (21198602+119861119862)] minus 2 (119860

2+119861119862)

sdot [(1198911119906119906119906 +1198912119906119906V) 1198612minus 2 (1198911119906119906V +1198912119906VV) 119860119861

minus1198912VVV119861119862]

(53)

Thus we have the following result

Theorem4 Assume that condition (18) is satisfied when 119899 = 0andC0(1205820) is defined by (53) Then the spatially homogeneousHopf bifurcation of system (1) at (1205820 0 0) is supercritical (respsubcritical) if

C0 (1205820)

1198601015840 (1205820) + 1198631015840 (1205820)lt 0 (119903119890119904119901 gt 0) (54)

Moreover if each eigenvalue of 119871119895(1205820) has negative real parts

for all 119895 isin N then the above spatially homogeneous bifurcatingperiodic solutions are stable (resp unstable) when

C0 (1205820) lt 0 (119903119890119904119901 gt 0) (55)

4 Spatially NonhomogeneousHopf Bifurcation

Notice that the spatially nonhomogeneous periodic solu-tions of (1) at (1205820 0 0) from Hopf bifurcation are unstableAccordingly in this section we will calculate Re 1198881(1205820) in (40)in order to determine the direction of Hopf bifurcation ofspatially nonhomogeneous periodic solutions of system (1) at(1205820 0 0) To this end we always assume that 119899 isin N in (18)throughout this section and still represent 119860(1205820) 119861(1205820) and119862(1205820) by119860 119861 and119862 respectivelyThus 119902lowast defined in (22) hasthe form

119902lowast= (

119886lowast119899

119887lowast119899

) cos 119899

ℓ119909

= (

1205960 + 119894 (119860 minus 11988911198992ℓ2)

ℓ1205871205960

minus119894119861

ℓ1205871205960

) cos 119899

ℓ119909

(56)

where

1205960 = radic(119860 minus1198891119899

2

ℓ2)(119863 minus

11988921198992

ℓ2) minus 119861119862

= radicminus(119860 minus1198891119899

2

ℓ2)

2

minus 119861119862

(57)

Since when 119899 isin N

intℓ120587

0cos3 119899

ℓ119909 119889119909 = 0 (58)

one can obtain

⟨119902lowast 119876119902119902⟩ = ⟨119902

lowast 119876119902119902⟩ = ⟨119902

lowast 119876119902119902⟩ = ⟨119902

lowast 119876119902119902⟩

= 0(59)

Thus in order to calculate Re 1198881(1205820) it remains to compute

⟨119902lowast 11987611990811119902

⟩

⟨119902lowast 11987611990820119902

⟩

⟨119902lowast 119862119902119902119902

⟩

(60)

Journal of Applied Mathematics 7

Let all the second- and third-order partial derivatives of119891119896(120582 119906 V) (119896 = 1 2) with respect to 119906 and V be evaluated at

(1205820 0 0) and let

119888119896119899

= 119891119896119906119906

+ 2119891119896119906V119887119899 +119891

119896VV1198872119899

119889119896119899

= 119891119896119906119906

+119891119896119906V (119887119899 + 119887

119899) +119891119896VV

100381610038161003816100381611988711989910038161003816100381610038162

119890119896119899

= 119891119896119906119906119906

+119891119896119906119906V (2119887119899 + 119887

119899) +119891119896119906VV (2

100381610038161003816100381611988711989910038161003816100381610038162+ 119887

2119899)

+119891119896VVV

100381610038161003816100381611988711989910038161003816100381610038162119887119899

119896 = 1 2

(61)

Then from (30) and (31) one can observe

119876119902119902

= (1198881119899

1198882119899) cos2 119899

ℓ119909 =

12(1198881119899

1198882119899)(1+ cos 2119899

ℓ119909)

119876119902119902

= (1198891119899

1198892119899) cos2 119899

ℓ119909 =

12(1198891119899

1198892119899)(1+ cos 2119899

ℓ119909)

119862119902119902119902

= (1198901119899

1198902119899) cos3 119899

ℓ119909

(62)

In view of (34) (59) and (62) we have

11986720 = 119876119902119902

=12(1198881119899

1198882119899)(1+ cos 2119899

ℓ119909)

11986711 = 119876119902119902

=12(1198891119899

1198892119899)(1+ cos 2119899

ℓ119909)

(63)

Equalities (63) show that the calculation of [21198941205960 minus 119871(1205820)]minus1

and [119871(1205820)]minus1 will be restricted on the subspaces spanned by

eigenmodes 1 and cos(2119899ℓ)119909Let

1205721 =(1211988911198892 minus 31198892

1) 1198994

ℓ4minus3 (1198892 minus 1198891) 1198601198992

ℓ2minus 31205962

0

1205722 =6 (1198891 + 1198892) 119899

21205960ℓ2

1205723 =11988921119899

4

ℓ4+

(1198892 minus 1198891) 1198601198992

ℓ2minus 31205962

0

1205724 = minus2 (1198891 + 1198892) 119899

21205960ℓ2

(64)

Then under condition (18) one can derive

[21198941205960119868 minus 1198712119899 (1205820)]minus1

=1

1205721 + 1198941205722

sdot(

21198941205960 + 119860 +(31198892 minus 1198891) 119899

2

ℓ2119861

119862 21198941205960 minus 119860 +41198891119899

2

ℓ2

)

[21198941205960119868 minus 1198710 (1205820)]minus1

=1

1205723 + 1198941205723

sdot(21198941205960 + 119860 minus

(1198891 + 1198892) 1198992

ℓ2119861

119862 21198941205960 minus 119860

)

(65)

From (36) and (61) we have

11990820 = [21198941205960119868 minus 1198712119899 (1205820)]

minus1

2cos 2119899

ℓ119909

+[21198941205960119868 minus 1198710 (1205820)]

minus1

2(

1198881119899

1198882119899) =

12 (1205721 + 1198941205722)

sdot(

[21198941205960 + 119860 +(31198892 minus 1198891) 119899

2

ℓ2] 1198881119899 + 1198611198882119899

1198621198881119899 + (21198941205960 minus 119860 +41198891119899

2

ℓ2) 1198882119899

)

sdot cos 2119899ℓ

119909 +1

2 (1205723 + 1198941205724)

sdot([21198941205960 + 119860 minus

(1198891 + 1198892) 1198992

ℓ2] 1198881119899 + 1198611198882119899

1198621198881119899 + (21198941205960 minus 119860) 1198882119899

)

(66)

Similarly we can get

11990811 =121205725

(

[119860 +(31198892 minus 1198891) 119899

2

ℓ2]1198891119899 + 1198611198892119899

1198621198891119899 + (41198891119899

2

ℓ2minus 119860)1198892119899

)

sdot cos 2119899ℓ

119909 +121205726

sdot([119860 minus

(1198891 + 1198892) 1198992

ℓ2]1198891119899 + 1198611198892119899

1198621198891119899 minus 1198601198892119899

)

(67)

where

1205725 =(1211988911198892 minus 31198892

1) 1198994

ℓ4minus3 (1198892 minus 1198891) 119860119899

2

ℓ2+120596

20

1205726 =11988921119899

4

ℓ4+

(1198892 minus 1198891) 1198601198992

ℓ2+120596

20

(68)

8 Journal of Applied Mathematics

From (30) we have

11987611990820119902

= (1198911119906119906120585 + 1198911119906V120578 + 1198911VV120574

1198912119906119906120585 + 1198912119906V120578 + 1198912VV120574) cos 119899

ℓ119909 cos 2119899

ℓ119909

+(1198911119906119906120591 + 1198911119906V120594 + 1198911VV120577

1198912119906119906120591 + 1198912119906V120594 + 1198912VV120577) cos 119899

ℓ119909

11987611990811119902

= (1198911119906119906120585 + 1198911119906V120578 + 1198911VV120574

1198912119906119906120585 + 1198912119906V120578 + 1198912VV120574) cos 119899

ℓ119909 cos 2119899

ℓ119909

+(1198911119906119906120591 + 1198911119906V120594 + 1198911VV120577

1198912119906119906120591 + 1198912119906V120594 + 1198912VV120577) cos 119899

ℓ119909

(69)

with

120585 =(21198941205960 + 119860 + (31198892 minus 1198891) 119899

2ℓ

2) 1198881119899 + 1198611198882119899

2 (1205721 + 1198941205722)

120578 =(21198941205960 + 119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860 minus 1198941205960) + 119861119862

2119861 (1205721 + 1198941205722)

sdot 1198881119899 +(51198891119899

2ℓ2 minus 2119860 + 1198941205960)

2 (1205721 + 1198941205722)1198882119899

120574 =[1198621198881119899 + (21198941205960 minus 119860 + 41198891119899

2ℓ2) 1198882119899] (11988911198992ℓ2 minus 119860 minus 1198941205960)

2119861 (1205721 + 1198941205722)

120591 =(21198941205960 + 119860 minus (1198891 + 1198892) 119899

2ℓ2) 1198881119899 + 1198611198882119899

2 (1205723 + 1198941205724)

120594 =(21198941205960 + 119860 minus (1198891 + 1198892) 119899

2ℓ2) (11988911198992ℓ2 minus 119860 minus 1198941205960) + 119861119862

2119861 (1205723 + 1198941205724)

sdot 1198881119899 +(1198891119899

2ℓ2 minus 2119860 + 1198941205960)

2 (1205723 + 1198941205724)1198882119899

120577 =[1198621198881119899 + (21198941205960 minus 119860) 1198882119899] (1198891119899

2ℓ2 minus 119860 minus 1198941205960)

2119861 (1205723 + 1198941205724)

120585 =(119860 + (31198892 minus 1198891) 119899

2ℓ2) 1198891119899 + 1198611198892119899

21205725

120578 =(119860 + (31198892 minus 1198891) 119899

2ℓ2) (1198941205960 minus 119860 + 11988911198992ℓ2) + 119861119862

211986112057251198891119899

+(1198941205960 minus 2119860 + 51198891119899

2ℓ2)

212057251198892119899

120574 =(1198941205960 minus 119860 + 1198891119899

2ℓ2) [1198621198891119899 + (411988911198992ℓ2 minus 119860) 1198892119899]

21198611205725

120591 =(119860 minus (1198891 + 1198892) 119899

2ℓ2) 1198891119899 + 1198611198892119899

21205726

120594 =(119860 minus (1198891 + 1198892) 119899

2ℓ2) (1198941205960 minus 119860 + 11988911198992ℓ2) + 119861119862

211986112057261198891119899

+(1198941205960 minus 2119860 + 1198891119899

2ℓ2)

212057261198892119899

120577 =(1198941205960 minus 119860 + 1198891119899

2ℓ2) (1198621198891119899 minus 1198601198892119899)

21198611205726

(70)

Notice that for 119899 isin N

intℓ120587

0cos2 119899

ℓ119909 119889119909 =

ℓ120587

2

intℓ120587

0cos 2119899

ℓ119909cos2 119899

ℓ119909 119889119909 =

ℓ120587

4

(71)

It follows from (69) that

⟨119902lowast 11987611990820119902

⟩ =ℓ120587

4[119886lowast

119899(1198911119906119906120585 +1198911119906V120578 +1198911VV120574)

+ 119887lowast

119899(1198912119906119906120585 +1198912119906V120578 +1198912VV120574)]

+ℓ120587

2[119886lowast

119899(1198911119906119906120591 +1198911119906V120594+1198911VV120577)

+ 119887lowast

119899(1198912119906119906120591 +1198912119906V120594+1198912VV120577)]

⟨119902lowast 11987611990811119902

⟩ =ℓ120587

4[119886lowast

119899(1198911119906119906120585 +1198911119906V120578 +1198911VV120574)

+ 119887lowast

119899(1198912119906119906120585 +1198912119906V120578 +1198912VV120574)]

+ℓ120587

2[119886lowast

119899(1198911119906119906120591 +1198911119906V120594+1198911VV120577)

+ 119887lowast

119899(1198912119906119906120591 +1198912119906V120594+1198912VV120577)]

(72)

Substituting 119887119899= (1198941205960minus119860(1205820)+1198891119899

2ℓ2)119861(1205820) into (61) gives

Re 119888119896119899

=119891119896119906119906

1198612 minus 2119891119896119906V (119860 minus 1198891119899

2ℓ2) 119861 + 119891119896VV [2 (119860 minus 1198891119899

2ℓ2)2+ 119861119862]

1198612

Im 119888119896119899

=21205960 [119891119896119906V119861 minus 119891

119896VV (119860 minus 11988911198992ℓ2)]

1198612

Journal of Applied Mathematics 9

119889119896119899

=119891119896119906119906

(119860 minus 11988911198992ℓ2) minus 2119891

119896119906V119861 minus 119891119896VV119862

119861

Re 119890119896119899

=119891119896119906119906119906

1198612 minus 3119891119896119906119906V (119860 minus 1198891119899

2ℓ2) 119861 + 2119891119896119906VV (119860 minus 1198891119899

2ℓ2)2+ 119891119896VVV (119860 minus 1198891119899

2ℓ2) 119862

1198612

Im 119890119896119899

=1205960 [119891119896119906119906V119861 minus 2119891

119896119906VV (119860 minus 11988911198992ℓ2) minus 119891

119896VVV119862]

1198612

119896 = 1 2(73)

Then from (70) and (73) we have

Re 120585 =[(119860 + (31198892 minus 1198891) 119899

2ℓ2)Re 1198881119899 + 119861Re 1198882119899 minus 21205960 Im 1198881119899] 1205721 + [(119860 + (31198892 minus 1198891) 119899

2ℓ2) Im 1198881119899 + 119861 Im 1198882119899 + 21205960 Re 1198881119899] 1205722

2 (12057221 + 1205722

2)

Im 120585 =[(119860 + (31198892 minus 1198891) 119899

2ℓ2) Im 1198881119899 + 119861 Im 1198882119899 + 21205960 Re 1198881119899] 1205721 minus [(119860 + (31198892 minus 1198891) 1198992ℓ2)Re 1198881119899 + 119861Re 1198882119899 minus 21205960 Im 1198881119899] 1205722

2 (12057221 + 1205722

2)

Re 120578

=[(119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0]Re 1198881119899 minus 31205960 ((1198891 minus 1198892) 1198992ℓ2 minus 119860) Im 1198881119899 + 119861 (51198891119899

2ℓ2 minus 2119860)Re 1198882119899 minus 1198611205960 Im 1198882119899

2119861 (12057221 + 1205722

2)

sdot 1205721

+[(119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0] Im 1198881119899 + 31205960 ((1198891 minus 1198892) 1198992ℓ2 minus 119860)Re 1198881119899 + 119861 (51198891119899

2ℓ2 minus 2119860) Im 1198882119899 + 1198611205960 Re 11988821198992119861 (1205722

1 + 12057222)

sdot 1205722

Im 120578

=[(119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0] Im 1198881119899 + 31205960 ((1198891 minus 1198892) 1198992ℓ2 minus 119860)Re 1198881119899 + 119861 (51198891119899

2ℓ2 minus 2119860) Im 1198882119899 + 1198611205960 Re 11988821198992119861 (1205722

1 + 12057222)

sdot 1205721

minus[(119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0]Re 1198881119899 minus 31205960 ((1198891 minus 1198892) 1198992ℓ2 minus 119860) Im 1198881119899 + 119861 (51198891119899

2ℓ2 minus 2119860)Re 1198882119899 minus 1198611205960 Im 1198882119899

2119861 (12057221 + 1205722

2)

sdot 1205722

Re 120574 =(1198891119899

2ℓ2 minus 119860) [119862Re 1198881119899 minus (119860 minus 411988911198992ℓ2)Re 1198882119899 minus 21205960 Im 1198882119899] + 1205960 [119862 Im 1198881119899 minus (119860 minus 41198891119899

2ℓ2) Im 1198882119899 + 21205960 Re 1198882119899]2119861 (1205722

1 + 12057222)

1205721

+(1198891119899

2ℓ2 minus 119860) [119862 Im 1198881119899 minus (119860 minus 411988911198992ℓ2) Im 1198882119899 + 21205960 Re 1198882119899] minus 1205960 [119862Re 1198881119899 minus (119860 minus 41198891119899

2ℓ2)Re 1198882119899 minus 21205960 Im 1198882119899]

2119861 (12057221 + 1205722

2)1205722

Im 120574 =(1198891119899

2ℓ2 minus 119860) [119862 Im 1198881119899 minus (119860 minus 411988911198992ℓ2) Im 1198882119899 + 21205960 Re 1198882119899] minus 1205960 [119862Re 1198881119899 minus (119860 minus 41198891119899

2ℓ2)Re 1198882119899 minus 21205960 Im 1198882119899]

2119861 (12057221 + 1205722

2)1205721

10 Journal of Applied Mathematics

minus(1198891119899

2ℓ2 minus 119860) [119862Re 1198881119899 minus (119860 minus 411988911198992ℓ2)Re 1198882119899 minus 21205960 Im 1198882119899] + 1205960 [119862 Im 1198881119899 minus (119860 minus 41198891119899

2ℓ2) Im 1198882119899 + 21205960 Re 1198882119899]2119861 (1205722

1 + 12057222)

1205722

Re 120591 =[(119860 + (1198891 + 1198892) 119899

2ℓ

2)Re 1198881119899 + 119861Re 1198882119899 minus 21205960 Im 1198881119899] 1205723 + [(119860 + (1198891 + 1198892) 119899

2ℓ

2) Im 1198881119899 + 119861 Im 1198882119899 + 21205960 Re 1198881119899] 1205724

2 (12057223 + 1205722

4)

Im 120591 =[(119860 + (1198891 + 1198892) 119899

2ℓ2) Im 1198881119899 + 119861 Im 1198882119899 + 21205960 Re 1198881119899] 1205723 minus [(119860 + (1198891 + 1198892) 1198992ℓ2)Re 1198881119899 + 119861Re 1198882119899 minus 21205960 Im 1198881119899] 1205724

2 (12057223 + 1205722

4)

Re120594

=[(119860 minus (1198891 + 1198892) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0]Re 1198881119899 minus 1205960 ((31198891 + 1198892) 1198992ℓ2 minus 3119860) Im 1198881119899 + 119861 (1198891119899

2ℓ2 minus 2119860)Re 1198882119899 minus 1198611205960 Im 1198882119899

2119861 (12057223 + 1205722

4)

sdot 1205723

+[(119860 minus (1198891 + 1198892) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0] Im 1198881119899 + 1205960 ((31198891 + 1198892) 1198992ℓ2 minus 3119860)Re 1198881119899 + 119861 (1198891119899

2ℓ2 minus 2119860) Im 1198882119899 + 1198611205960 Re 11988821198992119861 (1205722

3 + 12057224)

sdot 1205724

Im120594

=[(119860 minus (1198891 + 1198892) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0] Im 1198881119899 + 1205960 ((31198891 + 1198892) 1198992ℓ2 minus 3119860)Re 1198881119899 + 119861 (1198891119899

2ℓ2 minus 2119860) Im 1198882119899 + 1198611205960 Re 11988821198992119861 (1205722

3 + 12057224)

sdot 1205723

minus[(119860 minus (1198891 + 1198892) 119899

2ℓ

2) (1198891119899

2ℓ

2minus 119860) + 119861119862 + 21205962

0]Re 1198881119899 minus 1205960 ((31198891 + 1198892) 1198992ℓ

2minus 3119860) Im 1198881119899 + 119861 (1198891119899

2ℓ

2minus 2119860)Re 1198882119899 minus 1198611205960 Im 1198882119899

2119861 (12057223 + 1205722

4)

sdot 1205724

Re 120577 =(1198891119899

2ℓ

2minus 119860) (119862Re 1198881119899 minus 119860Re 1198882119899 minus 21205960 Im 1198882119899) + 1205960 (119862 Im 1198881119899 minus 119860 Im 1198882119899 + 21205960 Re 1198882119899)

2119861 (12057223 + 1205722

4)1205723

+(1198891119899

2ℓ2 minus 119860) (119862 Im 1198881119899 minus 119860 Im 1198882119899 + 21205960 Re 1198882119899) minus 1205960 (119862Re 1198881119899 minus 119860Re 1198882119899 minus 21205960 Im 1198882119899)

2119861 (12057223 + 1205722

4)1205724

Im 120577 =(1198891119899

2ℓ2 minus 119860) (119862 Im 1198881119899 minus 119860 Im 1198882119899 + 21205960 Re 1198882119899) minus 1205960 (119862Re 1198881119899 minus 119860Re 1198882119899 minus 21205960 Im 1198882119899)

2119861 (12057223 + 1205722

4)1205723

minus(1198891119899

2ℓ2 minus 119860) (119862Re 1198881119899 minus 119860Re 1198882119899 minus 21205960 Im 1198882119899) + 1205960 (119862 Im 1198881119899 minus 119860 Im 1198882119899 + 21205960 Re 1198882119899)2119861 (1205722

3 + 12057224)

1205724

Re 120578 =[(119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862] 1198891119899 + 119861 (51198891119899

2ℓ2 minus 2119860)1198892119899

21198611205725

Re 120578 =1205960 [(119860 + (31198892 minus 1198891) 119899

2ℓ2) 1198891119899 + 1198611198892119899]

21205725

Re 120574 =(1198891119899

2ℓ2 minus 119860) [1198621198891119899 + (411988911198992ℓ2 minus 119860) 1198892119899]

21198611205725

Im 120574 =1205960 [1198621198891119899 + (41198891119899

2ℓ2 minus 119860) 1198892119899]

21198611205725

Re120594 =[(119860 minus (1198891 + 1198892) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862] 1198891119899 + 119861 (1198891119899

2ℓ2 minus 2119860)1198892119899

21198611205726

Im120594 =1205960 [(119860 minus (1198891 + 1198892) 119899

2ℓ2) 1198891119899 + 1198611198892119899]

21205726119861 (1205820)

Re 120577 =(1198621198891119899 minus 1198601198892119899) (1198891119899

2ℓ

2minus 119860)

21198611205726

Im 120577 =1205960 (1198621198891119899 minus 1198601198892119899)

21198611205726

(74)

Journal of Applied Mathematics 11

Since ℓ120587119886lowast

119899= 1 minus 119894(119860 minus 1198891119899

2ℓ2)1205960 and ℓ120587119887lowast

119899= minus1198941198611205960

one can get

Re ⟨119902lowast 11987611990820119902

⟩ =14[1198911119906119906 (Re 120585 + 2 Re 120591)

+1198911119906V (Re 120578 + 2 Re120594) +1198911VV (Re 120574 + 2 Re 120577)]

+(119860 minus 1198891119899

2ℓ2)

41205960[1198911119906119906 (Im 120585 + 2 Im 120591)

+1198911119906V (Im 120578 + 2 Im120594) +1198911VV (Im 120574 + 2 Im 120577)]

+119861

41205960[1198912119906119906 (Im 120585 + 2 Im 120591)

+1198912119906V (Im 120578 + 2 Im120594) +1198912VV (Im 120574 + 2 Im 120577)]

Re ⟨119902lowast 11987611990811119902

⟩ =14[1198911119906119906 (120585 + 2120591)

+1198911119906V (Re 120578 + 2 Re120594) +1198911VV (Re 120574 + 2 Re 120577)]

+(119860 minus 1198891119899

2ℓ2)

41205960[1198911119906119906 (120585 + 2120591)

+1198911119906V (Re 120578 + 2 Re120594) +1198911VV (Re 120574 + 2 Re 120577)]

+119861

41205960[1198912119906V (Im 120578 + 2 Im120594)

+1198912VV (Im 120574 + 2 Im 120577)]

(75)

In addition it follows from intℓ120587

0 cos4(119899ℓ)119909 119889119909 = 3ℓ1205878and (62) that

⟨119902lowast 119862119902119902119902

⟩ =3ℓ1205878

(119886lowast

1198991198901119899 + 119887

lowast

1198991198902119899) (76)

Therefore

Re ⟨119902lowast 119862119902119902119902

⟩ =381198612 1198911119906119906119906119861

2

minus 31198911119906119906V (119860minus1198891119899

2

ℓ2)119861+ 21198911119906VV (119860minus

11988911198992

ℓ2)

2

+1198911VVV (119860minus1198891119899

2

ℓ2)119862+(119860minus

11988911198992

ℓ2)

sdot [1198911119906119906V119861minus 21198911119906VV (119860minus1198891119899

2

ℓ2)minus1198911VVV119862]

+119861[1198912119906119906V119861minus 21198912119906VV (119860minus1198891119899

2

ℓ2)minus1198912VVV119862]

(77)

Now by (40) we have

Re 1198881 (1205820) = Re ⟨119902lowast 11987611990811119902

⟩+12Re ⟨119902lowast 119876

11990820119902⟩+

12

sdotRe ⟨119902lowast 119862119902119902119902

⟩

=18[1198911119906119906 (Re 120585 + 2 Re 120591 + 2120585 + 4120591)

+1198911119906V (Re 120578 + 2 Re120594+ 2 Re 120578 + 4 Re120594)

+1198911VV (Re 120574 + 2 Re 120577 + 2 Re 120574 + 4 Re 120577)]

+(119860 minus 1198891119899

2ℓ2)

81205960[1198911119906119906 (Im 120585 + 2 Im 120591 + 2120585 + 4120591)

+1198911119906V (Im 120578 + 2 Im120594+ 2 Re 120578 + 4 Re120594)

+1198911VV (Im 120574 + 2 Im 120577 + 2 Re 120574 + 4 Re 120577)]

+119861

81205960[1198912119906119906 (Im 120585 + 2 Im 120591)

+1198912119906V (Im 120578119868+ 2 Im120594+ 2 Im 120578 + 4 Im120594)

+1198912VV (Im 120574 + Im 120577 + 2 Im 120574 + 4 Im 120577)]

+3

161198612 11989111199061199061199061198612minus 31198911119906119906V (119860minus

11988911198992

ℓ2)119861

+ 21198911119906VV (119860minus1198891119899

2

ℓ2)

2

+1198911VVV (119860minus1198891119899

2

ℓ2)119862

+(119860minus1198891119899

2

ℓ2)

sdot [1198911119906119906V119861minus 21198911119906VV (119860minus1198891119899

2

ℓ2)minus1198911VVV119862]

+119861[1198912119906119906V119861minus 21198912119906VV (119860minus1198891119899

2

ℓ2)minus1198912VVV119862]

(78)

Thus we have the following result

Theorem 5 Assume that condition (18) holds for 119899 isin N Thenthe spatially nonhomogeneous Hopf bifurcation of system (1) at(1205820 0 0) is supercritical (resp subcritical) if

Re 1198881 (1205820)

1198601015840 (1205820) + 1198631015840 (1205820)lt 0 (119903119890119904119901 gt 0) (79)

here Re 1198881(1205820) is given by (78)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

12 Journal of Applied Mathematics

Acknowledgments

Cun-Hua Zhang was supported by the Natural ScienceFoundation of Gansu Province (1212RJZA065) Xiang-PingYan was supported by the National Natural Science Foun-dation of China (11261028) Gansu Province Natural ScienceFoundation (145RJZA216) and China Scholarship Council

References

[1] A Andronov E A Leontovich I I Gordon and A G MaierTheory of Bifurcations of Dynamical Systems on a Plane IsraelProgram for Scientific Translations Jerusalem Israel 1971

[2] J Hale Theory of Functional Differential Equations SpringerNew York NY USA 1977

[3] B D Hassard N D Kazarinoff and Y H Wan Theory andApplications of Hopf Bifurcation Cambridge University PressCambridge UK 1981

[4] H Kielhofer Bifurcation Theory An Introduction with Appli-cations to PDEs vol 156 of Applied Mathematical SciencesSpringer New York NY USA 2004

[5] YA KuznetsovElements of Applied BifurcationTheory AppliedMathematical Sciences Springer New York NY USA 1998

[6] J-H Wu Theory and Applications of Partial Functional Differ-ential Equations Springer New York NY USA 1996

[7] F Yi JWei and J Shi ldquoBifurcation and spatiotemporal patternsin a homogeneous diffusive predator-prey systemrdquo Journal ofDifferential Equations vol 246 no 5 pp 1944ndash1977 2009

[8] L-J Yang and X-W Zeng ldquoStability of singular Hopf bifurca-tionsrdquo Journal of Differential Equations vol 206 no 1 pp 30ndash54 2004

[9] J Y Jin J P Shi J J Wei and F Q Yi ldquoBifurcations of patternedsolutions in the diffusive Lengyel-Epstein system of CIMAchemical reactionsrdquo Rocky Mountain Journal of Mathematicsvol 43 no 5 pp 1637ndash1674 2013

[10] S G Ruan ldquoDiffusion-driven instability in the Gierer-Mein-hardtmodel ofmorphogenesisrdquoNatural ResourceModeling vol11 no 2 pp 131ndash142 1998

[11] F Yi J Wei and J Shi ldquoDiffusion-driven instability andbifurcation in the Lengyel-Epstein systemrdquo Nonlinear AnalysisReal World Applications vol 9 no 3 pp 1038ndash1051 2008

[12] F Q Yi J J Wei and J P Shi ldquoGlobal asymptotical behavior ofthe Lengyel-Epstein reaction-diffusion systemrdquoApplied Mathe-matics Letters vol 22 no 1 pp 52ndash55 2009

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Page 3: Research Article Normal Forms of Hopf Bifurcation for a ...

Journal of Applied Mathematics 3

where 119886119895 119887119895isin C (119895 isin N0) Then (10) can be written into

infin

sum119895=0

119871119895(120582)(

119886119895

119887119895

) cos119895

ℓ119909 = 120573 (120582)

infin

sum119895=0

(119886119895

119887119895

) cos119895

ℓ119909 (12)

From the orthogonality of the function sequencecos(119895ℓ)119909infin

119895=0 one can get from (12) that for each 119895 isin N0

119871119895(120582)(

119886119895

119887119895

) = 120573 (120582)(119886119895

119887119895

) (13)

Since (120601 120595) isin 119883C is the eigenfunction of 119871(120582) correspondingto the eigenvalue 120573(120582) (120601 120595) = 0 and so there must be some119899 isin N0 such that 0 = (119886

119899 119887119899) isin C times C Therefore 120573(120582) is the

eigenvalue of the matrix 119871119899

If 120573(120582) is the eigenvalue of some matrix 119871119899 then there

exists a nonzero vector (119886119899 119887119899) isin C times C such that (13) holds

Let

(120601

120595) = (

119886119899

119887119899

) cos 119899

ℓ119909 (14)

Then (120601 120595) = 0 and

119871 (120582) (120601

120595) = 119871 (120582) (

119886119899

119887119899

) cos 119899

ℓ119909 = 119871

119899(119886119899

119887119899

) cos 119899

ℓ119909

= 120582(119886119899

119887119899

) cos 119899

ℓ119909 = 120573 (120582) (

120601

120595)

(15)

This demonstrates that 120573(120582) is an eigenvalue of 119871(120582) and thusthe proof is complete

Lemma 1 shows that under assumption (H) there isa unique 119899 isin N0 such that plusmn1198941205960 are purely imaginaryeigenvalues of 119871

119899(1205820) that is 119879119899(1205820) = 0 and 119863

119899(1205820) gt 0

Furthermore it is easy to see that 119879119895(1205820) = 0 for any 119895 = 119899

Therefore 119871119895(1205820) (119895 = 119899) has eigenvalues with zero real parts

if and only if119863119895(1205820) = 0 Assume that 120573(120582) = 120572(120582) + 119894120596(120582) is

the eigenvalue of 119871(120582) for 120582 sufficiently approaching 1205820Thenby the smoothness of 119891

119896(119896 = 1 2) we know that 120573(120582) is also

the eigenvalue of 119871119899(120582) namely 120573(120582) satisfies the following

equation

1205732minus120573119879119899(120582) +119863

119899(120582) = 0 (16)

Under the assumption (H) differentiating the above equationwith respect to 120582 at 1205820 yields

119889120572 (1205820)

119889120582=12[1198601015840(1205820) +119863

1015840(1205820)] (17)

Based on the above discussion condition (H) has thefollowing equivalent form

119879119899(1205820) = 0

119863119899(1205820) gt 0

1198601015840(1205820) +119863

1015840(1205820) = 0

for some 119899 isin N0 119863119895(1205820) = 0 for any 119895 isin N0

(18)

Then we know that 1205960 = radic119863119899(1205820) and 119861(1205820) 119862(1205820) cannot

be equal to zero simultaneously when the hypothesis (H) issatisfied Therefore the eigenvector of 119871

119899(1205820) corresponding

to the eigenvalue 1198941205960 can be chosen as

(119886119899

119887119899

) = (

1

1198941205960 minus 119860 (1205820) + 11988911198992ℓ2

119861 (1205820)

) (19)

and thus the eigenfunction of 119871(1205820) corresponding to the

eigenvalue 1198941205960 has the form

119902 = (119886119899

119887119899

) cos 119899

ℓ119909

= (

1

1198941205960 minus 119860 (1205820) + 11988911198992ℓ2

119861 (1205820)

) cos 119899

ℓ119909

(20)

Let the linear operator 119871lowast(1205820) with the domain 119863119871lowast(1205820)

=

119883C be defined by

119871lowast(1205820) = (

11988911205972

1205971199092 + 119860 (1205820) 119862 (1205820)

119861 (1205820) 11988921205972

1205971199092 + 119863 (1205820)

) (21)

Then 119871lowast(1205820) is the adjoint operator of the operator119871(1205820) suchthat ⟨119880 119871(1205820)119881⟩ = ⟨119871lowast(1205820)119880 119881⟩ with 119880119881 isin 119883C Similarto the choice of the eigenfunction 119902 of the operator 119871(1205820)corresponding to the eigenvalue 1198941205960 we can choose

119902lowast= (

119886lowast119899

119887lowast119899

) cos 119899

ℓ119909

= (

1205960 + 119894 (119860 (1205820) minus 11988911198992ℓ2)

21205960 intℓ120587

0 cos2 (119899ℓ) 119909 119889119909

minus119894119861 (1205820)

21205960 intℓ120587

0 cos2 (119899ℓ) 119909 119889119909

) cos 119899

ℓ119909

(22)

such that119871lowast(1205820) 119902

lowast= minus 1198941205960119902

⟨119902lowast 119902⟩ = 1

⟨119902lowast 119902⟩ = 0

(23)

Define 119883119862 and 119883119878 by 119883119862 = 119911119902 + 119911119902 | 119911 isin C and 119883119878 = 119880 isin

119883 | ⟨119902lowast 119880⟩ = 0 respectively Then119883 can be decomposed asthe direct sum of 119883119862 and 119883119878 that is 119883 = 119883119862 oplus 119883119878 Thus forany 119880 = (119906 V) isin 119883 there exists 119911 isin C and 119908 = (1199081 1199082) isin 119883

119904

such that119880 = 119911119902+ 119911 119902 +119908

or

119906 = 119911119886119899cos 119899

ℓ119909 + 119911 119886

119899cos 119899

ℓ119909 + 1199081

V = 119911119887119899cos 119899

ℓ119909 + 119911119887

119899cos 119899

ℓ119909 + 1199082

(24)

4 Journal of Applied Mathematics

Define 119865(120582 119880) by

119865 (120582 119880) = (1198911 (120582 119906 V) minus 119860 (120582) 119906 minus 119861 (120582) V

1198912 (120582 119906 V) minus 119862 (120582) 119906 minus 119863 (120582) V) (25)

Then system (1) can be rewritten into the following abstractform

119889119880

119889119905= 119871 (120582)119880+119865 (120582 119880) (26)

When 120582 = 1205820 system (26) is reduced to

119889119880

119889119905= 119871 (1205820) 119880+1198650 (119880) (27)

where 1198650(119880) = 119865(120582 119880)|120582=1205820

In terms of (23) and decomposi-tion (24) system (27) can be transformed into the followingsystem in (119911 119908) coordinates

119889119911

119889119905= 1198941205960119911 + ⟨119902

lowast 1198650⟩

119889119908

119889119905= 119871 (1205820) 119908+119867 (119911 119911 119908)

(28)

where

119867(119911 119911 119908) = 1198650 minus ⟨119902lowast 1198650⟩ 119902 minus ⟨119902

lowast 1198650⟩ 119902

1198650 = 1198650 (119911119902 + 119911 119902 +119908) (29)

For 119883 = (1199091 1199092) 119884 = (1199101 1199102) and 119885 = (1199111 1199112) isin

119883C define the symmetric multilinear forms 119876(119883 119884) and119862(119883 119884 119885) respectively by

119876 (119883 119884) = (

2sum119896119895=1

12059721198911 (1205820 1205851 1205852)

120597120585119896120597120585119895

10038161003816100381610038161003816100381610038161003816100381610038161205851=1205852=0119909119896119910119895

2sum119896119895=1

12059721198912 (1205820 1205851 1205852)

120597120585119896120597120585119895

10038161003816100381610038161003816100381610038161003816100381610038161205851=1205852=0119909119896119910119895

) (30)

119862 (119883 119884 119885)

= (

2sum119896119895119897=1

12059731198911 (1205820 1205851 1205852)

120597120585119896120597120585119895120597120585119897

10038161003816100381610038161003816100381610038161003816100381610038161205851=1205852=0119909119896119910119895119911119897

2sum119896119895119897=1

12059731198912 (1205820 1205851 1205852)

120597120585119896120597120585119895120597120585119897

10038161003816100381610038161003816100381610038161003816100381610038161205851=1205852=0119909119896119910119895119911119897

)(31)

Then for 119880 = (119906 V) isin 119883 we have

1198650 (119880) =12119876 (119880119880) +

16119862 (119880119880119880) +119874 (|119880|

4) (32)

For the simplicity of notations we will use 119876119883119884

and 119862119883119884119885

todenote 119876(119883 119884) and 119862(119883 119884 119885) respectively

Let

119867(119911 119911 119908) =119867202

1199112+11986711119911119911 +

119867022

1199112+119874 (|119911|

3)

+119874 (|119911| |119908|)

(33)

Then from (29) and (32) one can get

11986720 = 119876119902119902

minus⟨119902lowast 119876119902119902⟩ 119902 minus ⟨119902

lowast 119876119902119902⟩ 119902

11986711 = 119876119902119902

minus⟨119902lowast 119876119902119902⟩ 119902 minus ⟨119902

lowast 119876119902119902⟩ 119902

(34)

From the center manifold theorem in [3] we can rewrite119908 inthe form

119908 =119908202

1199112+11990811119911119911 +

119908022

1199112+119874 (|119911|

3) (35)

The second equation of (28) (33) and (35) yields

11990820 = [21198941205960119868 minus 119871 (1205820)]minus1

11986720

11990811 = minus [119871 (1205820)]minus1

11986711(36)

Substituting (35) into the first equation of (28) gives theequation of reaction-diffusion system (1) restricted on thecenter manifold at (1205820 0 0) as

119889119911

119889119905= 1198941205960119911 + sum

2le119896+119895le3

119892119896119895

119896119895119911119896119911119895+119874 (|119911|

4) (37)

where 11989220 = ⟨119902lowast 119876119902119902⟩ 11989211 = ⟨119902lowast 119876

119902119902⟩ 11989202 = ⟨119902lowast 119876

119902 119902⟩ and

11989221 = 2 ⟨119902lowast 11987611990811119902

⟩+⟨119902lowast 11987611990820119902

⟩+⟨119902lowast 119862119902119902119902

⟩ (38)

The dynamics of (28) can be determined by the dynamics of(37)

In addition it can be observed from [3] that when 120582

approaches sufficiently 1205820 the Poincare normal form of (26)has the form

= (120572 (120582) + 119894120596 (120582)) 119911 + 119911

119872

sum119895=1

119888119895(120582) (119911119911)

119895 (39)

where 119911 is a complex variable119872 ge 1 and 119888119895(120582) are complex-

valued coefficients with

1198881 (1205820) =119894

21205960(1198922011989211 minus 2 100381610038161003816100381611989211

10038161003816100381610038162minus13100381610038161003816100381611989202

10038161003816100381610038162)+

119892212

=119894

21205960(⟨119902lowast 119876119902119902⟩ ⟨119902lowast 119876119902119902⟩minus 2 10038161003816100381610038161003816

⟨119902lowast 119876119902119902⟩10038161003816100381610038161003816

2

minus1310038161003816100381610038161003816⟨119902lowast 119876119902 119902

⟩10038161003816100381610038161003816

2)+⟨119902

lowast 11987611990811119902

⟩+12⟨119902lowast 11987611990820119902

⟩

+12⟨119902lowast 119862119902119902119902

⟩

(40)

The direction of Hopf bifurcation and the stability of thebifurcating periodic solutions of (1) at (1205820 0 0) can bedetermined by the sign of Re 1198881(1205820) andwe have the followingconclusion

Theorem 2 Assume that condition (H) (or equivalently (18))holdsThen system (1) undergoes a supercritical (or subcritical)Hopf bifurcation at (0 0) when 120582 = 1205820 if

11205721015840 (1205820)

Re 1198881 (1205820) lt 0 (119903119890119904119901 gt 0) (41)

In addition if all other eigenvalues of 119871(1205820) have negative realparts then the bifurcating periodic solutions are stable (respunstable) when Re 1198881(1205820) lt 0 (119903119890119904119901 gt 0)

Journal of Applied Mathematics 5

3 Spatially Homogeneous Hopf Bifurcation

From the description in the previous section we know thatHopf bifurcation of (1) at (1205820 0 0) is spatially homogeneousif condition (18) holds when 119899 = 0 In the present section wecompute Re 1198881(1205820) in (40) in order to determine the directionof spatially homogeneous Hopf bifurcation and the stabilityof bifurcating periodic solutions of (1) at (1205820 0 0) followingthe algorithm described in this pervious section

Lemma 3 If condition (18) is satisfied when 119899 = 0 then11986720 =

11986711 = 0

Proof From (20) and (22) one can see

1198860 = 1

1198870 =1198941205960 minus 119860 (1205820)

119861 (1205820)

119886lowast

0 =1205960 + 119894119860 (1205820)

2ℓ1205871205960

119887lowast

0 = minus119894119861 (1205820)

2ℓ1205871205960

(42)

where

1205960 = radic119860 (1205820)119863 (1205820) minus 119861 (1205820) 119862 (1205820)

= radicminus1198602 (1205820) minus 119861 (1205820) 119862 (1205820)

(43)

Let all the partial derivatives of 119891119896(120582 119906 V) (119896 = 1 2) be

evaluated at (1205820 0 0) and let 1198881198960 1198891198960 and 119890

1198960 (119896 = 1 2) bedefined respectively by

1198881198960 = 119891

119896119906119906+ 2119891119896119906V1198870 +119891

119896VV11988720

1198891198960 = 119891

119896119906119906+119891119896119906V (1198870 + 1198870) +119891

119896VV10038161003816100381610038161198870

10038161003816100381610038162

1198901198960 = 119891

119896119906119906119906+119891119896119906119906V (21198870 + 1198870) +119891

119896119906VV (210038161003816100381610038161198870

10038161003816100381610038162+ 119887

20 )

+119891119896VVV

10038161003816100381610038161198870100381610038161003816100381621198870

(44)

Then from (30) and (31) we can get

119876119902119902

= (11988810

11988820)

119876119902119902

= (11988910

11988920)

119862119902119902119902

= (11989010

11989020)

(45)

Therefore

⟨119902lowast 119876119902119902⟩ = int

ℓ120587

0(119886lowast

0 11988810 + 119887lowast

0 11988820)

= ℓ120587 (119886lowast

0 11988810 + 119887lowast

0 11988820)

⟨119902lowast 119876119902119902⟩ = int

ℓ120587

0(119886lowast

0 11988810 + 119887lowast

0 11988820)

= ℓ120587 (119886lowast

0 11988810 + 119887lowast

0 11988820)

⟨119902lowast 119876119902119902⟩ = int

ℓ120587

0(119886lowast

011988910 + 119887lowast

011988920)

= ℓ120587 (119886lowast

011988910 + 119887lowast

011988920)

⟨119902lowast 119876119902119902⟩ = int

ℓ120587

0(119886lowast

0 11988910 + 119887lowast

0 11988920)

= ℓ120587 (119886lowast

0 11988910 + 119887lowast

0 11988920)

(46)

From (34) and (46) one can obtain

11986720

= (

11988810 minus ℓ120587 [(119886lowast0 + 119886lowast

0 ) 11988810 + (119887lowast0 + 119887lowast

0) 11988820]

11988820 minus ℓ120587 [(119886lowast

01198870 + 119886lowast0 1198870) 11988810 + (119887lowast

01198870 + 119887lowast0 1198870) 11988820]

)

11986711

= (

11988910 minus ℓ120587 [(119886lowast0 + 119886lowast

0 ) 11988910 + (119887lowast0 + 119887lowast

0) 11988920]

11988920 minus ℓ120587 [(119886lowast

01198870 + 119886lowast0 1198870) 11988910 + (119887lowast

01198870 + 119887lowast0 1198870) 11988920]

)

(47)

Notice from (42) that

119886lowast

0 + 119886lowast

0 = 119887lowast

01198870 + 119887lowast

0 1198870 =1ℓ120587

119887lowast

0 + 119887lowast

0 = 119886lowast

01198870 + 119886lowast

0 1198870 = 0(48)

The conclusion follows by substituting (48) into (47)

Lemma 3 and (36) imply 11990820 = 11990811 = 0 when 119899 = 0 in(18) and thus we have

11989221 = ⟨119902lowast 119862119902119902119902

⟩ = intℓ120587

0(119886lowast

0 11989010 + 119887lowast

0 11989020)

= ℓ120587 (119886lowast

0 11989010 + 119887lowast

0 11989020)

(49)

It follows from (40) that

2 Re 1198881 (1205820) = Re( 119894

1205960⟨119902lowast 119876119902119902⟩ ⟨119902lowast 119876119902119902⟩

+⟨119902lowast 119862119902119902119902

⟩)

6 Journal of Applied Mathematics

= Re[

[

119894ℓ120587 (119886lowast

0 11988810 + 119887lowast

0 11988820) ℓ120587 (119886lowast

011988910 + 119887lowast

011988920)

1205960

+ ℓ120587 (119886lowast

0 11989010 + 119887lowast

0 11989020)]

]

(50)

We represent 119860(1205820) 119861(1205820) and 119862(1205820) by 119860 119861 and119862 respectively for the simplicity of notations and underassumption (18) with 119899 = 0 substituting 1198870 in (42) into (44)yields that for 119896 = 1 2

Re 1198881198960 =

119891119896119906119906

1198612 minus 2119891119896119906V119860119861 + 119891

119896VV (21198602 + 119861119862)

1198612

Im 1198881198960 = 21205960

119891119896119906V119861 minus 119891

119896VV119860

1198612

1198891198960 =

119891119896119906119906

119860 minus 2119891119896119906V119861 minus 119891

119896VV119862

119861

Re 1198901198960 =

119891119896119906119906119906

1198612minus 3119891119896119906119906V119860119861 + 2119891

119896119906VV1198602+ 119891119896VVV119860119862

1198612

Im 1198901198960 = 1205960

119891119896119906119906V119861 minus 2119891

119896119906VV119860 minus 119891119896VVV119862

1198612

(51)

From (42) (50) and (51) one can derive

2 Re 1198881 (1205820) =C0 (1205820)

412059620119861

2 (52)

where

C0 (1205820) = [11989111199061199061198602+ (1198912119906119906 minus 21198911119906V) 119860119861minus1198911VV119860119862

minus 21198912119906V1198612minus1198912VV119861119862] (1198911119906119906119861+1198911VV119862+ 21198912119906V119861

minus 21198912VV119860)+ (1198911119906119906119860minus 21198911119906V119861minus1198911VV119862) [1198911119906119906119860119861

+ 21198911119906V119861119862minus1198911VV119860119862+11989121199061199061198612minus 21198912119906V119860119861

+1198912VV (21198602+119861119862)] minus 2 (119860

2+119861119862)

sdot [(1198911119906119906119906 +1198912119906119906V) 1198612minus 2 (1198911119906119906V +1198912119906VV) 119860119861

minus1198912VVV119861119862]

(53)

Thus we have the following result

Theorem4 Assume that condition (18) is satisfied when 119899 = 0andC0(1205820) is defined by (53) Then the spatially homogeneousHopf bifurcation of system (1) at (1205820 0 0) is supercritical (respsubcritical) if

C0 (1205820)

1198601015840 (1205820) + 1198631015840 (1205820)lt 0 (119903119890119904119901 gt 0) (54)

Moreover if each eigenvalue of 119871119895(1205820) has negative real parts

for all 119895 isin N then the above spatially homogeneous bifurcatingperiodic solutions are stable (resp unstable) when

C0 (1205820) lt 0 (119903119890119904119901 gt 0) (55)

4 Spatially NonhomogeneousHopf Bifurcation

Notice that the spatially nonhomogeneous periodic solu-tions of (1) at (1205820 0 0) from Hopf bifurcation are unstableAccordingly in this section we will calculate Re 1198881(1205820) in (40)in order to determine the direction of Hopf bifurcation ofspatially nonhomogeneous periodic solutions of system (1) at(1205820 0 0) To this end we always assume that 119899 isin N in (18)throughout this section and still represent 119860(1205820) 119861(1205820) and119862(1205820) by119860 119861 and119862 respectivelyThus 119902lowast defined in (22) hasthe form

119902lowast= (

119886lowast119899

119887lowast119899

) cos 119899

ℓ119909

= (

1205960 + 119894 (119860 minus 11988911198992ℓ2)

ℓ1205871205960

minus119894119861

ℓ1205871205960

) cos 119899

ℓ119909

(56)

where

1205960 = radic(119860 minus1198891119899

2

ℓ2)(119863 minus

11988921198992

ℓ2) minus 119861119862

= radicminus(119860 minus1198891119899

2

ℓ2)

2

minus 119861119862

(57)

Since when 119899 isin N

intℓ120587

0cos3 119899

ℓ119909 119889119909 = 0 (58)

one can obtain

⟨119902lowast 119876119902119902⟩ = ⟨119902

lowast 119876119902119902⟩ = ⟨119902

lowast 119876119902119902⟩ = ⟨119902

lowast 119876119902119902⟩

= 0(59)

Thus in order to calculate Re 1198881(1205820) it remains to compute

⟨119902lowast 11987611990811119902

⟩

⟨119902lowast 11987611990820119902

⟩

⟨119902lowast 119862119902119902119902

⟩

(60)

Journal of Applied Mathematics 7

Let all the second- and third-order partial derivatives of119891119896(120582 119906 V) (119896 = 1 2) with respect to 119906 and V be evaluated at

(1205820 0 0) and let

119888119896119899

= 119891119896119906119906

+ 2119891119896119906V119887119899 +119891

119896VV1198872119899

119889119896119899

= 119891119896119906119906

+119891119896119906V (119887119899 + 119887

119899) +119891119896VV

100381610038161003816100381611988711989910038161003816100381610038162

119890119896119899

= 119891119896119906119906119906

+119891119896119906119906V (2119887119899 + 119887

119899) +119891119896119906VV (2

100381610038161003816100381611988711989910038161003816100381610038162+ 119887

2119899)

+119891119896VVV

100381610038161003816100381611988711989910038161003816100381610038162119887119899

119896 = 1 2

(61)

Then from (30) and (31) one can observe

119876119902119902

= (1198881119899

1198882119899) cos2 119899

ℓ119909 =

12(1198881119899

1198882119899)(1+ cos 2119899

ℓ119909)

119876119902119902

= (1198891119899

1198892119899) cos2 119899

ℓ119909 =

12(1198891119899

1198892119899)(1+ cos 2119899

ℓ119909)

119862119902119902119902

= (1198901119899

1198902119899) cos3 119899

ℓ119909

(62)

In view of (34) (59) and (62) we have

11986720 = 119876119902119902

=12(1198881119899

1198882119899)(1+ cos 2119899

ℓ119909)

11986711 = 119876119902119902

=12(1198891119899

1198892119899)(1+ cos 2119899

ℓ119909)

(63)

Equalities (63) show that the calculation of [21198941205960 minus 119871(1205820)]minus1

and [119871(1205820)]minus1 will be restricted on the subspaces spanned by

eigenmodes 1 and cos(2119899ℓ)119909Let

1205721 =(1211988911198892 minus 31198892

1) 1198994

ℓ4minus3 (1198892 minus 1198891) 1198601198992

ℓ2minus 31205962

0

1205722 =6 (1198891 + 1198892) 119899

21205960ℓ2

1205723 =11988921119899

4

ℓ4+

(1198892 minus 1198891) 1198601198992

ℓ2minus 31205962

0

1205724 = minus2 (1198891 + 1198892) 119899

21205960ℓ2

(64)

Then under condition (18) one can derive

[21198941205960119868 minus 1198712119899 (1205820)]minus1

=1

1205721 + 1198941205722

sdot(

21198941205960 + 119860 +(31198892 minus 1198891) 119899

2

ℓ2119861

119862 21198941205960 minus 119860 +41198891119899

2

ℓ2

)

[21198941205960119868 minus 1198710 (1205820)]minus1

=1

1205723 + 1198941205723

sdot(21198941205960 + 119860 minus

(1198891 + 1198892) 1198992

ℓ2119861

119862 21198941205960 minus 119860

)

(65)

From (36) and (61) we have

11990820 = [21198941205960119868 minus 1198712119899 (1205820)]

minus1

2cos 2119899

ℓ119909

+[21198941205960119868 minus 1198710 (1205820)]

minus1

2(

1198881119899

1198882119899) =

12 (1205721 + 1198941205722)

sdot(

[21198941205960 + 119860 +(31198892 minus 1198891) 119899

2

ℓ2] 1198881119899 + 1198611198882119899

1198621198881119899 + (21198941205960 minus 119860 +41198891119899

2

ℓ2) 1198882119899

)

sdot cos 2119899ℓ

119909 +1

2 (1205723 + 1198941205724)

sdot([21198941205960 + 119860 minus

(1198891 + 1198892) 1198992

ℓ2] 1198881119899 + 1198611198882119899

1198621198881119899 + (21198941205960 minus 119860) 1198882119899

)

(66)

Similarly we can get

11990811 =121205725

(

[119860 +(31198892 minus 1198891) 119899

2

ℓ2]1198891119899 + 1198611198892119899

1198621198891119899 + (41198891119899

2

ℓ2minus 119860)1198892119899

)

sdot cos 2119899ℓ

119909 +121205726

sdot([119860 minus

(1198891 + 1198892) 1198992

ℓ2]1198891119899 + 1198611198892119899

1198621198891119899 minus 1198601198892119899

)

(67)

where

1205725 =(1211988911198892 minus 31198892

1) 1198994

ℓ4minus3 (1198892 minus 1198891) 119860119899

2

ℓ2+120596

20

1205726 =11988921119899

4

ℓ4+

(1198892 minus 1198891) 1198601198992

ℓ2+120596

20

(68)

8 Journal of Applied Mathematics

From (30) we have

11987611990820119902

= (1198911119906119906120585 + 1198911119906V120578 + 1198911VV120574

1198912119906119906120585 + 1198912119906V120578 + 1198912VV120574) cos 119899

ℓ119909 cos 2119899

ℓ119909

+(1198911119906119906120591 + 1198911119906V120594 + 1198911VV120577

1198912119906119906120591 + 1198912119906V120594 + 1198912VV120577) cos 119899

ℓ119909

11987611990811119902

= (1198911119906119906120585 + 1198911119906V120578 + 1198911VV120574

1198912119906119906120585 + 1198912119906V120578 + 1198912VV120574) cos 119899

ℓ119909 cos 2119899

ℓ119909

+(1198911119906119906120591 + 1198911119906V120594 + 1198911VV120577

1198912119906119906120591 + 1198912119906V120594 + 1198912VV120577) cos 119899

ℓ119909

(69)

with

120585 =(21198941205960 + 119860 + (31198892 minus 1198891) 119899

2ℓ

2) 1198881119899 + 1198611198882119899

2 (1205721 + 1198941205722)

120578 =(21198941205960 + 119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860 minus 1198941205960) + 119861119862

2119861 (1205721 + 1198941205722)

sdot 1198881119899 +(51198891119899

2ℓ2 minus 2119860 + 1198941205960)

2 (1205721 + 1198941205722)1198882119899

120574 =[1198621198881119899 + (21198941205960 minus 119860 + 41198891119899

2ℓ2) 1198882119899] (11988911198992ℓ2 minus 119860 minus 1198941205960)

2119861 (1205721 + 1198941205722)

120591 =(21198941205960 + 119860 minus (1198891 + 1198892) 119899

2ℓ2) 1198881119899 + 1198611198882119899

2 (1205723 + 1198941205724)

120594 =(21198941205960 + 119860 minus (1198891 + 1198892) 119899

2ℓ2) (11988911198992ℓ2 minus 119860 minus 1198941205960) + 119861119862

2119861 (1205723 + 1198941205724)

sdot 1198881119899 +(1198891119899

2ℓ2 minus 2119860 + 1198941205960)

2 (1205723 + 1198941205724)1198882119899

120577 =[1198621198881119899 + (21198941205960 minus 119860) 1198882119899] (1198891119899

2ℓ2 minus 119860 minus 1198941205960)

2119861 (1205723 + 1198941205724)

120585 =(119860 + (31198892 minus 1198891) 119899

2ℓ2) 1198891119899 + 1198611198892119899

21205725

120578 =(119860 + (31198892 minus 1198891) 119899

2ℓ2) (1198941205960 minus 119860 + 11988911198992ℓ2) + 119861119862

211986112057251198891119899

+(1198941205960 minus 2119860 + 51198891119899

2ℓ2)

212057251198892119899

120574 =(1198941205960 minus 119860 + 1198891119899

2ℓ2) [1198621198891119899 + (411988911198992ℓ2 minus 119860) 1198892119899]

21198611205725

120591 =(119860 minus (1198891 + 1198892) 119899

2ℓ2) 1198891119899 + 1198611198892119899

21205726

120594 =(119860 minus (1198891 + 1198892) 119899

2ℓ2) (1198941205960 minus 119860 + 11988911198992ℓ2) + 119861119862

211986112057261198891119899

+(1198941205960 minus 2119860 + 1198891119899

2ℓ2)

212057261198892119899

120577 =(1198941205960 minus 119860 + 1198891119899

2ℓ2) (1198621198891119899 minus 1198601198892119899)

21198611205726

(70)

Notice that for 119899 isin N

intℓ120587

0cos2 119899

ℓ119909 119889119909 =

ℓ120587

2

intℓ120587

0cos 2119899

ℓ119909cos2 119899

ℓ119909 119889119909 =

ℓ120587

4

(71)

It follows from (69) that

⟨119902lowast 11987611990820119902

⟩ =ℓ120587

4[119886lowast

119899(1198911119906119906120585 +1198911119906V120578 +1198911VV120574)

+ 119887lowast

119899(1198912119906119906120585 +1198912119906V120578 +1198912VV120574)]

+ℓ120587

2[119886lowast

119899(1198911119906119906120591 +1198911119906V120594+1198911VV120577)

+ 119887lowast

119899(1198912119906119906120591 +1198912119906V120594+1198912VV120577)]

⟨119902lowast 11987611990811119902

⟩ =ℓ120587

4[119886lowast

119899(1198911119906119906120585 +1198911119906V120578 +1198911VV120574)

+ 119887lowast

119899(1198912119906119906120585 +1198912119906V120578 +1198912VV120574)]

+ℓ120587

2[119886lowast

119899(1198911119906119906120591 +1198911119906V120594+1198911VV120577)

+ 119887lowast

119899(1198912119906119906120591 +1198912119906V120594+1198912VV120577)]

(72)

Substituting 119887119899= (1198941205960minus119860(1205820)+1198891119899

2ℓ2)119861(1205820) into (61) gives

Re 119888119896119899

=119891119896119906119906

1198612 minus 2119891119896119906V (119860 minus 1198891119899

2ℓ2) 119861 + 119891119896VV [2 (119860 minus 1198891119899

2ℓ2)2+ 119861119862]

1198612

Im 119888119896119899

=21205960 [119891119896119906V119861 minus 119891

119896VV (119860 minus 11988911198992ℓ2)]

1198612

Journal of Applied Mathematics 9

119889119896119899

=119891119896119906119906

(119860 minus 11988911198992ℓ2) minus 2119891

119896119906V119861 minus 119891119896VV119862

119861

Re 119890119896119899

=119891119896119906119906119906

1198612 minus 3119891119896119906119906V (119860 minus 1198891119899

2ℓ2) 119861 + 2119891119896119906VV (119860 minus 1198891119899

2ℓ2)2+ 119891119896VVV (119860 minus 1198891119899

2ℓ2) 119862

1198612

Im 119890119896119899

=1205960 [119891119896119906119906V119861 minus 2119891

119896119906VV (119860 minus 11988911198992ℓ2) minus 119891

119896VVV119862]

1198612

119896 = 1 2(73)

Then from (70) and (73) we have

Re 120585 =[(119860 + (31198892 minus 1198891) 119899

2ℓ2)Re 1198881119899 + 119861Re 1198882119899 minus 21205960 Im 1198881119899] 1205721 + [(119860 + (31198892 minus 1198891) 119899

2ℓ2) Im 1198881119899 + 119861 Im 1198882119899 + 21205960 Re 1198881119899] 1205722

2 (12057221 + 1205722

2)

Im 120585 =[(119860 + (31198892 minus 1198891) 119899

2ℓ2) Im 1198881119899 + 119861 Im 1198882119899 + 21205960 Re 1198881119899] 1205721 minus [(119860 + (31198892 minus 1198891) 1198992ℓ2)Re 1198881119899 + 119861Re 1198882119899 minus 21205960 Im 1198881119899] 1205722

2 (12057221 + 1205722

2)

Re 120578

=[(119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0]Re 1198881119899 minus 31205960 ((1198891 minus 1198892) 1198992ℓ2 minus 119860) Im 1198881119899 + 119861 (51198891119899

2ℓ2 minus 2119860)Re 1198882119899 minus 1198611205960 Im 1198882119899

2119861 (12057221 + 1205722

2)

sdot 1205721

+[(119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0] Im 1198881119899 + 31205960 ((1198891 minus 1198892) 1198992ℓ2 minus 119860)Re 1198881119899 + 119861 (51198891119899

2ℓ2 minus 2119860) Im 1198882119899 + 1198611205960 Re 11988821198992119861 (1205722

1 + 12057222)

sdot 1205722

Im 120578

=[(119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0] Im 1198881119899 + 31205960 ((1198891 minus 1198892) 1198992ℓ2 minus 119860)Re 1198881119899 + 119861 (51198891119899

2ℓ2 minus 2119860) Im 1198882119899 + 1198611205960 Re 11988821198992119861 (1205722

1 + 12057222)

sdot 1205721

minus[(119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0]Re 1198881119899 minus 31205960 ((1198891 minus 1198892) 1198992ℓ2 minus 119860) Im 1198881119899 + 119861 (51198891119899

2ℓ2 minus 2119860)Re 1198882119899 minus 1198611205960 Im 1198882119899

2119861 (12057221 + 1205722

2)

sdot 1205722

Re 120574 =(1198891119899

2ℓ2 minus 119860) [119862Re 1198881119899 minus (119860 minus 411988911198992ℓ2)Re 1198882119899 minus 21205960 Im 1198882119899] + 1205960 [119862 Im 1198881119899 minus (119860 minus 41198891119899

2ℓ2) Im 1198882119899 + 21205960 Re 1198882119899]2119861 (1205722

1 + 12057222)

1205721

+(1198891119899

2ℓ2 minus 119860) [119862 Im 1198881119899 minus (119860 minus 411988911198992ℓ2) Im 1198882119899 + 21205960 Re 1198882119899] minus 1205960 [119862Re 1198881119899 minus (119860 minus 41198891119899

2ℓ2)Re 1198882119899 minus 21205960 Im 1198882119899]

2119861 (12057221 + 1205722

2)1205722

Im 120574 =(1198891119899

2ℓ2 minus 119860) [119862 Im 1198881119899 minus (119860 minus 411988911198992ℓ2) Im 1198882119899 + 21205960 Re 1198882119899] minus 1205960 [119862Re 1198881119899 minus (119860 minus 41198891119899

2ℓ2)Re 1198882119899 minus 21205960 Im 1198882119899]

2119861 (12057221 + 1205722

2)1205721

10 Journal of Applied Mathematics

minus(1198891119899

2ℓ2 minus 119860) [119862Re 1198881119899 minus (119860 minus 411988911198992ℓ2)Re 1198882119899 minus 21205960 Im 1198882119899] + 1205960 [119862 Im 1198881119899 minus (119860 minus 41198891119899

2ℓ2) Im 1198882119899 + 21205960 Re 1198882119899]2119861 (1205722

1 + 12057222)

1205722

Re 120591 =[(119860 + (1198891 + 1198892) 119899

2ℓ

2)Re 1198881119899 + 119861Re 1198882119899 minus 21205960 Im 1198881119899] 1205723 + [(119860 + (1198891 + 1198892) 119899

2ℓ

2) Im 1198881119899 + 119861 Im 1198882119899 + 21205960 Re 1198881119899] 1205724

2 (12057223 + 1205722

4)

Im 120591 =[(119860 + (1198891 + 1198892) 119899

2ℓ2) Im 1198881119899 + 119861 Im 1198882119899 + 21205960 Re 1198881119899] 1205723 minus [(119860 + (1198891 + 1198892) 1198992ℓ2)Re 1198881119899 + 119861Re 1198882119899 minus 21205960 Im 1198881119899] 1205724

2 (12057223 + 1205722

4)

Re120594

=[(119860 minus (1198891 + 1198892) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0]Re 1198881119899 minus 1205960 ((31198891 + 1198892) 1198992ℓ2 minus 3119860) Im 1198881119899 + 119861 (1198891119899

2ℓ2 minus 2119860)Re 1198882119899 minus 1198611205960 Im 1198882119899

2119861 (12057223 + 1205722

4)

sdot 1205723

+[(119860 minus (1198891 + 1198892) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0] Im 1198881119899 + 1205960 ((31198891 + 1198892) 1198992ℓ2 minus 3119860)Re 1198881119899 + 119861 (1198891119899

2ℓ2 minus 2119860) Im 1198882119899 + 1198611205960 Re 11988821198992119861 (1205722

3 + 12057224)

sdot 1205724

Im120594

=[(119860 minus (1198891 + 1198892) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0] Im 1198881119899 + 1205960 ((31198891 + 1198892) 1198992ℓ2 minus 3119860)Re 1198881119899 + 119861 (1198891119899

2ℓ2 minus 2119860) Im 1198882119899 + 1198611205960 Re 11988821198992119861 (1205722

3 + 12057224)

sdot 1205723

minus[(119860 minus (1198891 + 1198892) 119899

2ℓ

2) (1198891119899

2ℓ

2minus 119860) + 119861119862 + 21205962

0]Re 1198881119899 minus 1205960 ((31198891 + 1198892) 1198992ℓ

2minus 3119860) Im 1198881119899 + 119861 (1198891119899

2ℓ

2minus 2119860)Re 1198882119899 minus 1198611205960 Im 1198882119899

2119861 (12057223 + 1205722

4)

sdot 1205724

Re 120577 =(1198891119899

2ℓ

2minus 119860) (119862Re 1198881119899 minus 119860Re 1198882119899 minus 21205960 Im 1198882119899) + 1205960 (119862 Im 1198881119899 minus 119860 Im 1198882119899 + 21205960 Re 1198882119899)

2119861 (12057223 + 1205722

4)1205723

+(1198891119899

2ℓ2 minus 119860) (119862 Im 1198881119899 minus 119860 Im 1198882119899 + 21205960 Re 1198882119899) minus 1205960 (119862Re 1198881119899 minus 119860Re 1198882119899 minus 21205960 Im 1198882119899)

2119861 (12057223 + 1205722

4)1205724

Im 120577 =(1198891119899

2ℓ2 minus 119860) (119862 Im 1198881119899 minus 119860 Im 1198882119899 + 21205960 Re 1198882119899) minus 1205960 (119862Re 1198881119899 minus 119860Re 1198882119899 minus 21205960 Im 1198882119899)

2119861 (12057223 + 1205722

4)1205723

minus(1198891119899

2ℓ2 minus 119860) (119862Re 1198881119899 minus 119860Re 1198882119899 minus 21205960 Im 1198882119899) + 1205960 (119862 Im 1198881119899 minus 119860 Im 1198882119899 + 21205960 Re 1198882119899)2119861 (1205722

3 + 12057224)

1205724

Re 120578 =[(119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862] 1198891119899 + 119861 (51198891119899

2ℓ2 minus 2119860)1198892119899

21198611205725

Re 120578 =1205960 [(119860 + (31198892 minus 1198891) 119899

2ℓ2) 1198891119899 + 1198611198892119899]

21205725

Re 120574 =(1198891119899

2ℓ2 minus 119860) [1198621198891119899 + (411988911198992ℓ2 minus 119860) 1198892119899]

21198611205725

Im 120574 =1205960 [1198621198891119899 + (41198891119899

2ℓ2 minus 119860) 1198892119899]

21198611205725

Re120594 =[(119860 minus (1198891 + 1198892) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862] 1198891119899 + 119861 (1198891119899

2ℓ2 minus 2119860)1198892119899

21198611205726

Im120594 =1205960 [(119860 minus (1198891 + 1198892) 119899

2ℓ2) 1198891119899 + 1198611198892119899]

21205726119861 (1205820)

Re 120577 =(1198621198891119899 minus 1198601198892119899) (1198891119899

2ℓ

2minus 119860)

21198611205726

Im 120577 =1205960 (1198621198891119899 minus 1198601198892119899)

21198611205726

(74)

Journal of Applied Mathematics 11

Since ℓ120587119886lowast

119899= 1 minus 119894(119860 minus 1198891119899

2ℓ2)1205960 and ℓ120587119887lowast

119899= minus1198941198611205960

one can get

Re ⟨119902lowast 11987611990820119902

⟩ =14[1198911119906119906 (Re 120585 + 2 Re 120591)

+1198911119906V (Re 120578 + 2 Re120594) +1198911VV (Re 120574 + 2 Re 120577)]

+(119860 minus 1198891119899

2ℓ2)

41205960[1198911119906119906 (Im 120585 + 2 Im 120591)

+1198911119906V (Im 120578 + 2 Im120594) +1198911VV (Im 120574 + 2 Im 120577)]

+119861

41205960[1198912119906119906 (Im 120585 + 2 Im 120591)

+1198912119906V (Im 120578 + 2 Im120594) +1198912VV (Im 120574 + 2 Im 120577)]

Re ⟨119902lowast 11987611990811119902

⟩ =14[1198911119906119906 (120585 + 2120591)

+1198911119906V (Re 120578 + 2 Re120594) +1198911VV (Re 120574 + 2 Re 120577)]

+(119860 minus 1198891119899

2ℓ2)

41205960[1198911119906119906 (120585 + 2120591)

+1198911119906V (Re 120578 + 2 Re120594) +1198911VV (Re 120574 + 2 Re 120577)]

+119861

41205960[1198912119906V (Im 120578 + 2 Im120594)

+1198912VV (Im 120574 + 2 Im 120577)]

(75)

In addition it follows from intℓ120587

0 cos4(119899ℓ)119909 119889119909 = 3ℓ1205878and (62) that

⟨119902lowast 119862119902119902119902

⟩ =3ℓ1205878

(119886lowast

1198991198901119899 + 119887

lowast

1198991198902119899) (76)

Therefore

Re ⟨119902lowast 119862119902119902119902

⟩ =381198612 1198911119906119906119906119861

2

minus 31198911119906119906V (119860minus1198891119899

2

ℓ2)119861+ 21198911119906VV (119860minus

11988911198992

ℓ2)

2

+1198911VVV (119860minus1198891119899

2

ℓ2)119862+(119860minus

11988911198992

ℓ2)

sdot [1198911119906119906V119861minus 21198911119906VV (119860minus1198891119899

2

ℓ2)minus1198911VVV119862]

+119861[1198912119906119906V119861minus 21198912119906VV (119860minus1198891119899

2

ℓ2)minus1198912VVV119862]

(77)

Now by (40) we have

Re 1198881 (1205820) = Re ⟨119902lowast 11987611990811119902

⟩+12Re ⟨119902lowast 119876

11990820119902⟩+

12

sdotRe ⟨119902lowast 119862119902119902119902

⟩

=18[1198911119906119906 (Re 120585 + 2 Re 120591 + 2120585 + 4120591)

+1198911119906V (Re 120578 + 2 Re120594+ 2 Re 120578 + 4 Re120594)

+1198911VV (Re 120574 + 2 Re 120577 + 2 Re 120574 + 4 Re 120577)]

+(119860 minus 1198891119899

2ℓ2)

81205960[1198911119906119906 (Im 120585 + 2 Im 120591 + 2120585 + 4120591)

+1198911119906V (Im 120578 + 2 Im120594+ 2 Re 120578 + 4 Re120594)

+1198911VV (Im 120574 + 2 Im 120577 + 2 Re 120574 + 4 Re 120577)]

+119861

81205960[1198912119906119906 (Im 120585 + 2 Im 120591)

+1198912119906V (Im 120578119868+ 2 Im120594+ 2 Im 120578 + 4 Im120594)

+1198912VV (Im 120574 + Im 120577 + 2 Im 120574 + 4 Im 120577)]

+3

161198612 11989111199061199061199061198612minus 31198911119906119906V (119860minus

11988911198992

ℓ2)119861

+ 21198911119906VV (119860minus1198891119899

2

ℓ2)

2

+1198911VVV (119860minus1198891119899

2

ℓ2)119862

+(119860minus1198891119899

2

ℓ2)

sdot [1198911119906119906V119861minus 21198911119906VV (119860minus1198891119899

2

ℓ2)minus1198911VVV119862]

+119861[1198912119906119906V119861minus 21198912119906VV (119860minus1198891119899

2

ℓ2)minus1198912VVV119862]

(78)

Thus we have the following result

Theorem 5 Assume that condition (18) holds for 119899 isin N Thenthe spatially nonhomogeneous Hopf bifurcation of system (1) at(1205820 0 0) is supercritical (resp subcritical) if

Re 1198881 (1205820)

1198601015840 (1205820) + 1198631015840 (1205820)lt 0 (119903119890119904119901 gt 0) (79)

here Re 1198881(1205820) is given by (78)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

12 Journal of Applied Mathematics

Acknowledgments

Cun-Hua Zhang was supported by the Natural ScienceFoundation of Gansu Province (1212RJZA065) Xiang-PingYan was supported by the National Natural Science Foun-dation of China (11261028) Gansu Province Natural ScienceFoundation (145RJZA216) and China Scholarship Council

References

[1] A Andronov E A Leontovich I I Gordon and A G MaierTheory of Bifurcations of Dynamical Systems on a Plane IsraelProgram for Scientific Translations Jerusalem Israel 1971

[2] J Hale Theory of Functional Differential Equations SpringerNew York NY USA 1977

[3] B D Hassard N D Kazarinoff and Y H Wan Theory andApplications of Hopf Bifurcation Cambridge University PressCambridge UK 1981

[4] H Kielhofer Bifurcation Theory An Introduction with Appli-cations to PDEs vol 156 of Applied Mathematical SciencesSpringer New York NY USA 2004

[5] YA KuznetsovElements of Applied BifurcationTheory AppliedMathematical Sciences Springer New York NY USA 1998

[6] J-H Wu Theory and Applications of Partial Functional Differ-ential Equations Springer New York NY USA 1996

[7] F Yi JWei and J Shi ldquoBifurcation and spatiotemporal patternsin a homogeneous diffusive predator-prey systemrdquo Journal ofDifferential Equations vol 246 no 5 pp 1944ndash1977 2009

[8] L-J Yang and X-W Zeng ldquoStability of singular Hopf bifurca-tionsrdquo Journal of Differential Equations vol 206 no 1 pp 30ndash54 2004

[9] J Y Jin J P Shi J J Wei and F Q Yi ldquoBifurcations of patternedsolutions in the diffusive Lengyel-Epstein system of CIMAchemical reactionsrdquo Rocky Mountain Journal of Mathematicsvol 43 no 5 pp 1637ndash1674 2013

[10] S G Ruan ldquoDiffusion-driven instability in the Gierer-Mein-hardtmodel ofmorphogenesisrdquoNatural ResourceModeling vol11 no 2 pp 131ndash142 1998

[11] F Yi J Wei and J Shi ldquoDiffusion-driven instability andbifurcation in the Lengyel-Epstein systemrdquo Nonlinear AnalysisReal World Applications vol 9 no 3 pp 1038ndash1051 2008

[12] F Q Yi J J Wei and J P Shi ldquoGlobal asymptotical behavior ofthe Lengyel-Epstein reaction-diffusion systemrdquoApplied Mathe-matics Letters vol 22 no 1 pp 52ndash55 2009

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 4: Research Article Normal Forms of Hopf Bifurcation for a ...

4 Journal of Applied Mathematics

Define 119865(120582 119880) by

119865 (120582 119880) = (1198911 (120582 119906 V) minus 119860 (120582) 119906 minus 119861 (120582) V

1198912 (120582 119906 V) minus 119862 (120582) 119906 minus 119863 (120582) V) (25)

Then system (1) can be rewritten into the following abstractform

119889119880

119889119905= 119871 (120582)119880+119865 (120582 119880) (26)

When 120582 = 1205820 system (26) is reduced to

119889119880

119889119905= 119871 (1205820) 119880+1198650 (119880) (27)

where 1198650(119880) = 119865(120582 119880)|120582=1205820

In terms of (23) and decomposi-tion (24) system (27) can be transformed into the followingsystem in (119911 119908) coordinates

119889119911

119889119905= 1198941205960119911 + ⟨119902

lowast 1198650⟩

119889119908

119889119905= 119871 (1205820) 119908+119867 (119911 119911 119908)

(28)

where

119867(119911 119911 119908) = 1198650 minus ⟨119902lowast 1198650⟩ 119902 minus ⟨119902

lowast 1198650⟩ 119902

1198650 = 1198650 (119911119902 + 119911 119902 +119908) (29)

For 119883 = (1199091 1199092) 119884 = (1199101 1199102) and 119885 = (1199111 1199112) isin

119883C define the symmetric multilinear forms 119876(119883 119884) and119862(119883 119884 119885) respectively by

119876 (119883 119884) = (

2sum119896119895=1

12059721198911 (1205820 1205851 1205852)

120597120585119896120597120585119895

10038161003816100381610038161003816100381610038161003816100381610038161205851=1205852=0119909119896119910119895

2sum119896119895=1

12059721198912 (1205820 1205851 1205852)

120597120585119896120597120585119895

10038161003816100381610038161003816100381610038161003816100381610038161205851=1205852=0119909119896119910119895

) (30)

119862 (119883 119884 119885)

= (

2sum119896119895119897=1

12059731198911 (1205820 1205851 1205852)

120597120585119896120597120585119895120597120585119897

10038161003816100381610038161003816100381610038161003816100381610038161205851=1205852=0119909119896119910119895119911119897

2sum119896119895119897=1

12059731198912 (1205820 1205851 1205852)

120597120585119896120597120585119895120597120585119897

10038161003816100381610038161003816100381610038161003816100381610038161205851=1205852=0119909119896119910119895119911119897

)(31)

Then for 119880 = (119906 V) isin 119883 we have

1198650 (119880) =12119876 (119880119880) +

16119862 (119880119880119880) +119874 (|119880|

4) (32)

For the simplicity of notations we will use 119876119883119884

and 119862119883119884119885

todenote 119876(119883 119884) and 119862(119883 119884 119885) respectively

Let

119867(119911 119911 119908) =119867202

1199112+11986711119911119911 +

119867022

1199112+119874 (|119911|

3)

+119874 (|119911| |119908|)

(33)

Then from (29) and (32) one can get

11986720 = 119876119902119902

minus⟨119902lowast 119876119902119902⟩ 119902 minus ⟨119902

lowast 119876119902119902⟩ 119902

11986711 = 119876119902119902

minus⟨119902lowast 119876119902119902⟩ 119902 minus ⟨119902

lowast 119876119902119902⟩ 119902

(34)

From the center manifold theorem in [3] we can rewrite119908 inthe form

119908 =119908202

1199112+11990811119911119911 +

119908022

1199112+119874 (|119911|

3) (35)

The second equation of (28) (33) and (35) yields

11990820 = [21198941205960119868 minus 119871 (1205820)]minus1

11986720

11990811 = minus [119871 (1205820)]minus1

11986711(36)

Substituting (35) into the first equation of (28) gives theequation of reaction-diffusion system (1) restricted on thecenter manifold at (1205820 0 0) as

119889119911

119889119905= 1198941205960119911 + sum

2le119896+119895le3

119892119896119895

119896119895119911119896119911119895+119874 (|119911|

4) (37)

where 11989220 = ⟨119902lowast 119876119902119902⟩ 11989211 = ⟨119902lowast 119876

119902119902⟩ 11989202 = ⟨119902lowast 119876

119902 119902⟩ and

11989221 = 2 ⟨119902lowast 11987611990811119902

⟩+⟨119902lowast 11987611990820119902

⟩+⟨119902lowast 119862119902119902119902

⟩ (38)

The dynamics of (28) can be determined by the dynamics of(37)

In addition it can be observed from [3] that when 120582

approaches sufficiently 1205820 the Poincare normal form of (26)has the form

= (120572 (120582) + 119894120596 (120582)) 119911 + 119911

119872

sum119895=1

119888119895(120582) (119911119911)

119895 (39)

where 119911 is a complex variable119872 ge 1 and 119888119895(120582) are complex-

valued coefficients with

1198881 (1205820) =119894

21205960(1198922011989211 minus 2 100381610038161003816100381611989211

10038161003816100381610038162minus13100381610038161003816100381611989202

10038161003816100381610038162)+

119892212

=119894

21205960(⟨119902lowast 119876119902119902⟩ ⟨119902lowast 119876119902119902⟩minus 2 10038161003816100381610038161003816

⟨119902lowast 119876119902119902⟩10038161003816100381610038161003816

2

minus1310038161003816100381610038161003816⟨119902lowast 119876119902 119902

⟩10038161003816100381610038161003816

2)+⟨119902

lowast 11987611990811119902

⟩+12⟨119902lowast 11987611990820119902

⟩

+12⟨119902lowast 119862119902119902119902

⟩

(40)

The direction of Hopf bifurcation and the stability of thebifurcating periodic solutions of (1) at (1205820 0 0) can bedetermined by the sign of Re 1198881(1205820) andwe have the followingconclusion

Theorem 2 Assume that condition (H) (or equivalently (18))holdsThen system (1) undergoes a supercritical (or subcritical)Hopf bifurcation at (0 0) when 120582 = 1205820 if

11205721015840 (1205820)

Re 1198881 (1205820) lt 0 (119903119890119904119901 gt 0) (41)

In addition if all other eigenvalues of 119871(1205820) have negative realparts then the bifurcating periodic solutions are stable (respunstable) when Re 1198881(1205820) lt 0 (119903119890119904119901 gt 0)

Journal of Applied Mathematics 5

3 Spatially Homogeneous Hopf Bifurcation

From the description in the previous section we know thatHopf bifurcation of (1) at (1205820 0 0) is spatially homogeneousif condition (18) holds when 119899 = 0 In the present section wecompute Re 1198881(1205820) in (40) in order to determine the directionof spatially homogeneous Hopf bifurcation and the stabilityof bifurcating periodic solutions of (1) at (1205820 0 0) followingthe algorithm described in this pervious section

Lemma 3 If condition (18) is satisfied when 119899 = 0 then11986720 =

11986711 = 0

Proof From (20) and (22) one can see

1198860 = 1

1198870 =1198941205960 minus 119860 (1205820)

119861 (1205820)

119886lowast

0 =1205960 + 119894119860 (1205820)

2ℓ1205871205960

119887lowast

0 = minus119894119861 (1205820)

2ℓ1205871205960

(42)

where

1205960 = radic119860 (1205820)119863 (1205820) minus 119861 (1205820) 119862 (1205820)

= radicminus1198602 (1205820) minus 119861 (1205820) 119862 (1205820)

(43)

Let all the partial derivatives of 119891119896(120582 119906 V) (119896 = 1 2) be

evaluated at (1205820 0 0) and let 1198881198960 1198891198960 and 119890

1198960 (119896 = 1 2) bedefined respectively by

1198881198960 = 119891

119896119906119906+ 2119891119896119906V1198870 +119891

119896VV11988720

1198891198960 = 119891

119896119906119906+119891119896119906V (1198870 + 1198870) +119891

119896VV10038161003816100381610038161198870

10038161003816100381610038162

1198901198960 = 119891

119896119906119906119906+119891119896119906119906V (21198870 + 1198870) +119891

119896119906VV (210038161003816100381610038161198870

10038161003816100381610038162+ 119887

20 )

+119891119896VVV

10038161003816100381610038161198870100381610038161003816100381621198870

(44)

Then from (30) and (31) we can get

119876119902119902

= (11988810

11988820)

119876119902119902

= (11988910

11988920)

119862119902119902119902

= (11989010

11989020)

(45)

Therefore

⟨119902lowast 119876119902119902⟩ = int

ℓ120587

0(119886lowast

0 11988810 + 119887lowast

0 11988820)

= ℓ120587 (119886lowast

0 11988810 + 119887lowast

0 11988820)

⟨119902lowast 119876119902119902⟩ = int

ℓ120587

0(119886lowast

0 11988810 + 119887lowast

0 11988820)

= ℓ120587 (119886lowast

0 11988810 + 119887lowast

0 11988820)

⟨119902lowast 119876119902119902⟩ = int

ℓ120587

0(119886lowast

011988910 + 119887lowast

011988920)

= ℓ120587 (119886lowast

011988910 + 119887lowast

011988920)

⟨119902lowast 119876119902119902⟩ = int

ℓ120587

0(119886lowast

0 11988910 + 119887lowast

0 11988920)

= ℓ120587 (119886lowast

0 11988910 + 119887lowast

0 11988920)

(46)

From (34) and (46) one can obtain

11986720

= (

11988810 minus ℓ120587 [(119886lowast0 + 119886lowast

0 ) 11988810 + (119887lowast0 + 119887lowast

0) 11988820]

11988820 minus ℓ120587 [(119886lowast

01198870 + 119886lowast0 1198870) 11988810 + (119887lowast

01198870 + 119887lowast0 1198870) 11988820]

)

11986711

= (

11988910 minus ℓ120587 [(119886lowast0 + 119886lowast

0 ) 11988910 + (119887lowast0 + 119887lowast

0) 11988920]

11988920 minus ℓ120587 [(119886lowast

01198870 + 119886lowast0 1198870) 11988910 + (119887lowast

01198870 + 119887lowast0 1198870) 11988920]

)

(47)

Notice from (42) that

119886lowast

0 + 119886lowast

0 = 119887lowast

01198870 + 119887lowast

0 1198870 =1ℓ120587

119887lowast

0 + 119887lowast

0 = 119886lowast

01198870 + 119886lowast

0 1198870 = 0(48)

The conclusion follows by substituting (48) into (47)

Lemma 3 and (36) imply 11990820 = 11990811 = 0 when 119899 = 0 in(18) and thus we have

11989221 = ⟨119902lowast 119862119902119902119902

⟩ = intℓ120587

0(119886lowast

0 11989010 + 119887lowast

0 11989020)

= ℓ120587 (119886lowast

0 11989010 + 119887lowast

0 11989020)

(49)

It follows from (40) that

2 Re 1198881 (1205820) = Re( 119894

1205960⟨119902lowast 119876119902119902⟩ ⟨119902lowast 119876119902119902⟩

+⟨119902lowast 119862119902119902119902

⟩)

6 Journal of Applied Mathematics

= Re[

[

119894ℓ120587 (119886lowast

0 11988810 + 119887lowast

0 11988820) ℓ120587 (119886lowast

011988910 + 119887lowast

011988920)

1205960

+ ℓ120587 (119886lowast

0 11989010 + 119887lowast

0 11989020)]

]

(50)

We represent 119860(1205820) 119861(1205820) and 119862(1205820) by 119860 119861 and119862 respectively for the simplicity of notations and underassumption (18) with 119899 = 0 substituting 1198870 in (42) into (44)yields that for 119896 = 1 2

Re 1198881198960 =

119891119896119906119906

1198612 minus 2119891119896119906V119860119861 + 119891

119896VV (21198602 + 119861119862)

1198612

Im 1198881198960 = 21205960

119891119896119906V119861 minus 119891

119896VV119860

1198612

1198891198960 =

119891119896119906119906

119860 minus 2119891119896119906V119861 minus 119891

119896VV119862

119861

Re 1198901198960 =

119891119896119906119906119906

1198612minus 3119891119896119906119906V119860119861 + 2119891

119896119906VV1198602+ 119891119896VVV119860119862

1198612

Im 1198901198960 = 1205960

119891119896119906119906V119861 minus 2119891

119896119906VV119860 minus 119891119896VVV119862

1198612

(51)

From (42) (50) and (51) one can derive

2 Re 1198881 (1205820) =C0 (1205820)

412059620119861

2 (52)

where

C0 (1205820) = [11989111199061199061198602+ (1198912119906119906 minus 21198911119906V) 119860119861minus1198911VV119860119862

minus 21198912119906V1198612minus1198912VV119861119862] (1198911119906119906119861+1198911VV119862+ 21198912119906V119861

minus 21198912VV119860)+ (1198911119906119906119860minus 21198911119906V119861minus1198911VV119862) [1198911119906119906119860119861

+ 21198911119906V119861119862minus1198911VV119860119862+11989121199061199061198612minus 21198912119906V119860119861

+1198912VV (21198602+119861119862)] minus 2 (119860

2+119861119862)

sdot [(1198911119906119906119906 +1198912119906119906V) 1198612minus 2 (1198911119906119906V +1198912119906VV) 119860119861

minus1198912VVV119861119862]

(53)

Thus we have the following result

Theorem4 Assume that condition (18) is satisfied when 119899 = 0andC0(1205820) is defined by (53) Then the spatially homogeneousHopf bifurcation of system (1) at (1205820 0 0) is supercritical (respsubcritical) if

C0 (1205820)

1198601015840 (1205820) + 1198631015840 (1205820)lt 0 (119903119890119904119901 gt 0) (54)

Moreover if each eigenvalue of 119871119895(1205820) has negative real parts

for all 119895 isin N then the above spatially homogeneous bifurcatingperiodic solutions are stable (resp unstable) when

C0 (1205820) lt 0 (119903119890119904119901 gt 0) (55)

4 Spatially NonhomogeneousHopf Bifurcation

Notice that the spatially nonhomogeneous periodic solu-tions of (1) at (1205820 0 0) from Hopf bifurcation are unstableAccordingly in this section we will calculate Re 1198881(1205820) in (40)in order to determine the direction of Hopf bifurcation ofspatially nonhomogeneous periodic solutions of system (1) at(1205820 0 0) To this end we always assume that 119899 isin N in (18)throughout this section and still represent 119860(1205820) 119861(1205820) and119862(1205820) by119860 119861 and119862 respectivelyThus 119902lowast defined in (22) hasthe form

119902lowast= (

119886lowast119899

119887lowast119899

) cos 119899

ℓ119909

= (

1205960 + 119894 (119860 minus 11988911198992ℓ2)

ℓ1205871205960

minus119894119861

ℓ1205871205960

) cos 119899

ℓ119909

(56)

where

1205960 = radic(119860 minus1198891119899

2

ℓ2)(119863 minus

11988921198992

ℓ2) minus 119861119862

= radicminus(119860 minus1198891119899

2

ℓ2)

2

minus 119861119862

(57)

Since when 119899 isin N

intℓ120587

0cos3 119899

ℓ119909 119889119909 = 0 (58)

one can obtain

⟨119902lowast 119876119902119902⟩ = ⟨119902

lowast 119876119902119902⟩ = ⟨119902

lowast 119876119902119902⟩ = ⟨119902

lowast 119876119902119902⟩

= 0(59)

Thus in order to calculate Re 1198881(1205820) it remains to compute

⟨119902lowast 11987611990811119902

⟩

⟨119902lowast 11987611990820119902

⟩

⟨119902lowast 119862119902119902119902

⟩

(60)

Journal of Applied Mathematics 7

Let all the second- and third-order partial derivatives of119891119896(120582 119906 V) (119896 = 1 2) with respect to 119906 and V be evaluated at

(1205820 0 0) and let

119888119896119899

= 119891119896119906119906

+ 2119891119896119906V119887119899 +119891

119896VV1198872119899

119889119896119899

= 119891119896119906119906

+119891119896119906V (119887119899 + 119887

119899) +119891119896VV

100381610038161003816100381611988711989910038161003816100381610038162

119890119896119899

= 119891119896119906119906119906

+119891119896119906119906V (2119887119899 + 119887

119899) +119891119896119906VV (2

100381610038161003816100381611988711989910038161003816100381610038162+ 119887

2119899)

+119891119896VVV

100381610038161003816100381611988711989910038161003816100381610038162119887119899

119896 = 1 2

(61)

Then from (30) and (31) one can observe

119876119902119902

= (1198881119899

1198882119899) cos2 119899

ℓ119909 =

12(1198881119899

1198882119899)(1+ cos 2119899

ℓ119909)

119876119902119902

= (1198891119899

1198892119899) cos2 119899

ℓ119909 =

12(1198891119899

1198892119899)(1+ cos 2119899

ℓ119909)

119862119902119902119902

= (1198901119899

1198902119899) cos3 119899

ℓ119909

(62)

In view of (34) (59) and (62) we have

11986720 = 119876119902119902

=12(1198881119899

1198882119899)(1+ cos 2119899

ℓ119909)

11986711 = 119876119902119902

=12(1198891119899

1198892119899)(1+ cos 2119899

ℓ119909)

(63)

Equalities (63) show that the calculation of [21198941205960 minus 119871(1205820)]minus1

and [119871(1205820)]minus1 will be restricted on the subspaces spanned by

eigenmodes 1 and cos(2119899ℓ)119909Let

1205721 =(1211988911198892 minus 31198892

1) 1198994

ℓ4minus3 (1198892 minus 1198891) 1198601198992

ℓ2minus 31205962

0

1205722 =6 (1198891 + 1198892) 119899

21205960ℓ2

1205723 =11988921119899

4

ℓ4+

(1198892 minus 1198891) 1198601198992

ℓ2minus 31205962

0

1205724 = minus2 (1198891 + 1198892) 119899

21205960ℓ2

(64)

Then under condition (18) one can derive

[21198941205960119868 minus 1198712119899 (1205820)]minus1

=1

1205721 + 1198941205722

sdot(

21198941205960 + 119860 +(31198892 minus 1198891) 119899

2

ℓ2119861

119862 21198941205960 minus 119860 +41198891119899

2

ℓ2

)

[21198941205960119868 minus 1198710 (1205820)]minus1

=1

1205723 + 1198941205723

sdot(21198941205960 + 119860 minus

(1198891 + 1198892) 1198992

ℓ2119861

119862 21198941205960 minus 119860

)

(65)

From (36) and (61) we have

11990820 = [21198941205960119868 minus 1198712119899 (1205820)]

minus1

2cos 2119899

ℓ119909

+[21198941205960119868 minus 1198710 (1205820)]

minus1

2(

1198881119899

1198882119899) =

12 (1205721 + 1198941205722)

sdot(

[21198941205960 + 119860 +(31198892 minus 1198891) 119899

2

ℓ2] 1198881119899 + 1198611198882119899

1198621198881119899 + (21198941205960 minus 119860 +41198891119899

2

ℓ2) 1198882119899

)

sdot cos 2119899ℓ

119909 +1

2 (1205723 + 1198941205724)

sdot([21198941205960 + 119860 minus

(1198891 + 1198892) 1198992

ℓ2] 1198881119899 + 1198611198882119899

1198621198881119899 + (21198941205960 minus 119860) 1198882119899

)

(66)

Similarly we can get

11990811 =121205725

(

[119860 +(31198892 minus 1198891) 119899

2

ℓ2]1198891119899 + 1198611198892119899

1198621198891119899 + (41198891119899

2

ℓ2minus 119860)1198892119899

)

sdot cos 2119899ℓ

119909 +121205726

sdot([119860 minus

(1198891 + 1198892) 1198992

ℓ2]1198891119899 + 1198611198892119899

1198621198891119899 minus 1198601198892119899

)

(67)

where

1205725 =(1211988911198892 minus 31198892

1) 1198994

ℓ4minus3 (1198892 minus 1198891) 119860119899

2

ℓ2+120596

20

1205726 =11988921119899

4

ℓ4+

(1198892 minus 1198891) 1198601198992

ℓ2+120596

20

(68)

8 Journal of Applied Mathematics

From (30) we have

11987611990820119902

= (1198911119906119906120585 + 1198911119906V120578 + 1198911VV120574

1198912119906119906120585 + 1198912119906V120578 + 1198912VV120574) cos 119899

ℓ119909 cos 2119899

ℓ119909

+(1198911119906119906120591 + 1198911119906V120594 + 1198911VV120577

1198912119906119906120591 + 1198912119906V120594 + 1198912VV120577) cos 119899

ℓ119909

11987611990811119902

= (1198911119906119906120585 + 1198911119906V120578 + 1198911VV120574

1198912119906119906120585 + 1198912119906V120578 + 1198912VV120574) cos 119899

ℓ119909 cos 2119899

ℓ119909

+(1198911119906119906120591 + 1198911119906V120594 + 1198911VV120577

1198912119906119906120591 + 1198912119906V120594 + 1198912VV120577) cos 119899

ℓ119909

(69)

with

120585 =(21198941205960 + 119860 + (31198892 minus 1198891) 119899

2ℓ

2) 1198881119899 + 1198611198882119899

2 (1205721 + 1198941205722)

120578 =(21198941205960 + 119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860 minus 1198941205960) + 119861119862

2119861 (1205721 + 1198941205722)

sdot 1198881119899 +(51198891119899

2ℓ2 minus 2119860 + 1198941205960)

2 (1205721 + 1198941205722)1198882119899

120574 =[1198621198881119899 + (21198941205960 minus 119860 + 41198891119899

2ℓ2) 1198882119899] (11988911198992ℓ2 minus 119860 minus 1198941205960)

2119861 (1205721 + 1198941205722)

120591 =(21198941205960 + 119860 minus (1198891 + 1198892) 119899

2ℓ2) 1198881119899 + 1198611198882119899

2 (1205723 + 1198941205724)

120594 =(21198941205960 + 119860 minus (1198891 + 1198892) 119899

2ℓ2) (11988911198992ℓ2 minus 119860 minus 1198941205960) + 119861119862

2119861 (1205723 + 1198941205724)

sdot 1198881119899 +(1198891119899

2ℓ2 minus 2119860 + 1198941205960)

2 (1205723 + 1198941205724)1198882119899

120577 =[1198621198881119899 + (21198941205960 minus 119860) 1198882119899] (1198891119899

2ℓ2 minus 119860 minus 1198941205960)

2119861 (1205723 + 1198941205724)

120585 =(119860 + (31198892 minus 1198891) 119899

2ℓ2) 1198891119899 + 1198611198892119899

21205725

120578 =(119860 + (31198892 minus 1198891) 119899

2ℓ2) (1198941205960 minus 119860 + 11988911198992ℓ2) + 119861119862

211986112057251198891119899

+(1198941205960 minus 2119860 + 51198891119899

2ℓ2)

212057251198892119899

120574 =(1198941205960 minus 119860 + 1198891119899

2ℓ2) [1198621198891119899 + (411988911198992ℓ2 minus 119860) 1198892119899]

21198611205725

120591 =(119860 minus (1198891 + 1198892) 119899

2ℓ2) 1198891119899 + 1198611198892119899

21205726

120594 =(119860 minus (1198891 + 1198892) 119899

2ℓ2) (1198941205960 minus 119860 + 11988911198992ℓ2) + 119861119862

211986112057261198891119899

+(1198941205960 minus 2119860 + 1198891119899

2ℓ2)

212057261198892119899

120577 =(1198941205960 minus 119860 + 1198891119899

2ℓ2) (1198621198891119899 minus 1198601198892119899)

21198611205726

(70)

Notice that for 119899 isin N

intℓ120587

0cos2 119899

ℓ119909 119889119909 =

ℓ120587

2

intℓ120587

0cos 2119899

ℓ119909cos2 119899

ℓ119909 119889119909 =

ℓ120587

4

(71)

It follows from (69) that

⟨119902lowast 11987611990820119902

⟩ =ℓ120587

4[119886lowast

119899(1198911119906119906120585 +1198911119906V120578 +1198911VV120574)

+ 119887lowast

119899(1198912119906119906120585 +1198912119906V120578 +1198912VV120574)]

+ℓ120587

2[119886lowast

119899(1198911119906119906120591 +1198911119906V120594+1198911VV120577)

+ 119887lowast

119899(1198912119906119906120591 +1198912119906V120594+1198912VV120577)]

⟨119902lowast 11987611990811119902

⟩ =ℓ120587

4[119886lowast

119899(1198911119906119906120585 +1198911119906V120578 +1198911VV120574)

+ 119887lowast

119899(1198912119906119906120585 +1198912119906V120578 +1198912VV120574)]

+ℓ120587

2[119886lowast

119899(1198911119906119906120591 +1198911119906V120594+1198911VV120577)

+ 119887lowast

119899(1198912119906119906120591 +1198912119906V120594+1198912VV120577)]

(72)

Substituting 119887119899= (1198941205960minus119860(1205820)+1198891119899

2ℓ2)119861(1205820) into (61) gives

Re 119888119896119899

=119891119896119906119906

1198612 minus 2119891119896119906V (119860 minus 1198891119899

2ℓ2) 119861 + 119891119896VV [2 (119860 minus 1198891119899

2ℓ2)2+ 119861119862]

1198612

Im 119888119896119899

=21205960 [119891119896119906V119861 minus 119891

119896VV (119860 minus 11988911198992ℓ2)]

1198612

Journal of Applied Mathematics 9

119889119896119899

=119891119896119906119906

(119860 minus 11988911198992ℓ2) minus 2119891

119896119906V119861 minus 119891119896VV119862

119861

Re 119890119896119899

=119891119896119906119906119906

1198612 minus 3119891119896119906119906V (119860 minus 1198891119899

2ℓ2) 119861 + 2119891119896119906VV (119860 minus 1198891119899

2ℓ2)2+ 119891119896VVV (119860 minus 1198891119899

2ℓ2) 119862

1198612

Im 119890119896119899

=1205960 [119891119896119906119906V119861 minus 2119891

119896119906VV (119860 minus 11988911198992ℓ2) minus 119891

119896VVV119862]

1198612

119896 = 1 2(73)

Then from (70) and (73) we have

Re 120585 =[(119860 + (31198892 minus 1198891) 119899

2ℓ2)Re 1198881119899 + 119861Re 1198882119899 minus 21205960 Im 1198881119899] 1205721 + [(119860 + (31198892 minus 1198891) 119899

2ℓ2) Im 1198881119899 + 119861 Im 1198882119899 + 21205960 Re 1198881119899] 1205722

2 (12057221 + 1205722

2)

Im 120585 =[(119860 + (31198892 minus 1198891) 119899

2ℓ2) Im 1198881119899 + 119861 Im 1198882119899 + 21205960 Re 1198881119899] 1205721 minus [(119860 + (31198892 minus 1198891) 1198992ℓ2)Re 1198881119899 + 119861Re 1198882119899 minus 21205960 Im 1198881119899] 1205722

2 (12057221 + 1205722

2)

Re 120578

=[(119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0]Re 1198881119899 minus 31205960 ((1198891 minus 1198892) 1198992ℓ2 minus 119860) Im 1198881119899 + 119861 (51198891119899

2ℓ2 minus 2119860)Re 1198882119899 minus 1198611205960 Im 1198882119899

2119861 (12057221 + 1205722

2)

sdot 1205721

+[(119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0] Im 1198881119899 + 31205960 ((1198891 minus 1198892) 1198992ℓ2 minus 119860)Re 1198881119899 + 119861 (51198891119899

2ℓ2 minus 2119860) Im 1198882119899 + 1198611205960 Re 11988821198992119861 (1205722

1 + 12057222)

sdot 1205722

Im 120578

=[(119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0] Im 1198881119899 + 31205960 ((1198891 minus 1198892) 1198992ℓ2 minus 119860)Re 1198881119899 + 119861 (51198891119899

2ℓ2 minus 2119860) Im 1198882119899 + 1198611205960 Re 11988821198992119861 (1205722

1 + 12057222)

sdot 1205721

minus[(119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0]Re 1198881119899 minus 31205960 ((1198891 minus 1198892) 1198992ℓ2 minus 119860) Im 1198881119899 + 119861 (51198891119899

2ℓ2 minus 2119860)Re 1198882119899 minus 1198611205960 Im 1198882119899

2119861 (12057221 + 1205722

2)

sdot 1205722

Re 120574 =(1198891119899

2ℓ2 minus 119860) [119862Re 1198881119899 minus (119860 minus 411988911198992ℓ2)Re 1198882119899 minus 21205960 Im 1198882119899] + 1205960 [119862 Im 1198881119899 minus (119860 minus 41198891119899

2ℓ2) Im 1198882119899 + 21205960 Re 1198882119899]2119861 (1205722

1 + 12057222)

1205721

+(1198891119899

2ℓ2 minus 119860) [119862 Im 1198881119899 minus (119860 minus 411988911198992ℓ2) Im 1198882119899 + 21205960 Re 1198882119899] minus 1205960 [119862Re 1198881119899 minus (119860 minus 41198891119899

2ℓ2)Re 1198882119899 minus 21205960 Im 1198882119899]

2119861 (12057221 + 1205722

2)1205722

Im 120574 =(1198891119899

2ℓ2 minus 119860) [119862 Im 1198881119899 minus (119860 minus 411988911198992ℓ2) Im 1198882119899 + 21205960 Re 1198882119899] minus 1205960 [119862Re 1198881119899 minus (119860 minus 41198891119899

2ℓ2)Re 1198882119899 minus 21205960 Im 1198882119899]

2119861 (12057221 + 1205722

2)1205721

10 Journal of Applied Mathematics

minus(1198891119899

2ℓ2 minus 119860) [119862Re 1198881119899 minus (119860 minus 411988911198992ℓ2)Re 1198882119899 minus 21205960 Im 1198882119899] + 1205960 [119862 Im 1198881119899 minus (119860 minus 41198891119899

2ℓ2) Im 1198882119899 + 21205960 Re 1198882119899]2119861 (1205722

1 + 12057222)

1205722

Re 120591 =[(119860 + (1198891 + 1198892) 119899

2ℓ

2)Re 1198881119899 + 119861Re 1198882119899 minus 21205960 Im 1198881119899] 1205723 + [(119860 + (1198891 + 1198892) 119899

2ℓ

2) Im 1198881119899 + 119861 Im 1198882119899 + 21205960 Re 1198881119899] 1205724

2 (12057223 + 1205722

4)

Im 120591 =[(119860 + (1198891 + 1198892) 119899

2ℓ2) Im 1198881119899 + 119861 Im 1198882119899 + 21205960 Re 1198881119899] 1205723 minus [(119860 + (1198891 + 1198892) 1198992ℓ2)Re 1198881119899 + 119861Re 1198882119899 minus 21205960 Im 1198881119899] 1205724

2 (12057223 + 1205722

4)

Re120594

=[(119860 minus (1198891 + 1198892) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0]Re 1198881119899 minus 1205960 ((31198891 + 1198892) 1198992ℓ2 minus 3119860) Im 1198881119899 + 119861 (1198891119899

2ℓ2 minus 2119860)Re 1198882119899 minus 1198611205960 Im 1198882119899

2119861 (12057223 + 1205722

4)

sdot 1205723

+[(119860 minus (1198891 + 1198892) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0] Im 1198881119899 + 1205960 ((31198891 + 1198892) 1198992ℓ2 minus 3119860)Re 1198881119899 + 119861 (1198891119899

2ℓ2 minus 2119860) Im 1198882119899 + 1198611205960 Re 11988821198992119861 (1205722

3 + 12057224)

sdot 1205724

Im120594

=[(119860 minus (1198891 + 1198892) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0] Im 1198881119899 + 1205960 ((31198891 + 1198892) 1198992ℓ2 minus 3119860)Re 1198881119899 + 119861 (1198891119899

2ℓ2 minus 2119860) Im 1198882119899 + 1198611205960 Re 11988821198992119861 (1205722

3 + 12057224)

sdot 1205723

minus[(119860 minus (1198891 + 1198892) 119899

2ℓ

2) (1198891119899

2ℓ

2minus 119860) + 119861119862 + 21205962

0]Re 1198881119899 minus 1205960 ((31198891 + 1198892) 1198992ℓ

2minus 3119860) Im 1198881119899 + 119861 (1198891119899

2ℓ

2minus 2119860)Re 1198882119899 minus 1198611205960 Im 1198882119899

2119861 (12057223 + 1205722

4)

sdot 1205724

Re 120577 =(1198891119899

2ℓ

2minus 119860) (119862Re 1198881119899 minus 119860Re 1198882119899 minus 21205960 Im 1198882119899) + 1205960 (119862 Im 1198881119899 minus 119860 Im 1198882119899 + 21205960 Re 1198882119899)

2119861 (12057223 + 1205722

4)1205723

+(1198891119899

2ℓ2 minus 119860) (119862 Im 1198881119899 minus 119860 Im 1198882119899 + 21205960 Re 1198882119899) minus 1205960 (119862Re 1198881119899 minus 119860Re 1198882119899 minus 21205960 Im 1198882119899)

2119861 (12057223 + 1205722

4)1205724

Im 120577 =(1198891119899

2ℓ2 minus 119860) (119862 Im 1198881119899 minus 119860 Im 1198882119899 + 21205960 Re 1198882119899) minus 1205960 (119862Re 1198881119899 minus 119860Re 1198882119899 minus 21205960 Im 1198882119899)

2119861 (12057223 + 1205722

4)1205723

minus(1198891119899

2ℓ2 minus 119860) (119862Re 1198881119899 minus 119860Re 1198882119899 minus 21205960 Im 1198882119899) + 1205960 (119862 Im 1198881119899 minus 119860 Im 1198882119899 + 21205960 Re 1198882119899)2119861 (1205722

3 + 12057224)

1205724

Re 120578 =[(119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862] 1198891119899 + 119861 (51198891119899

2ℓ2 minus 2119860)1198892119899

21198611205725

Re 120578 =1205960 [(119860 + (31198892 minus 1198891) 119899

2ℓ2) 1198891119899 + 1198611198892119899]

21205725

Re 120574 =(1198891119899

2ℓ2 minus 119860) [1198621198891119899 + (411988911198992ℓ2 minus 119860) 1198892119899]

21198611205725

Im 120574 =1205960 [1198621198891119899 + (41198891119899

2ℓ2 minus 119860) 1198892119899]

21198611205725

Re120594 =[(119860 minus (1198891 + 1198892) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862] 1198891119899 + 119861 (1198891119899

2ℓ2 minus 2119860)1198892119899

21198611205726

Im120594 =1205960 [(119860 minus (1198891 + 1198892) 119899

2ℓ2) 1198891119899 + 1198611198892119899]

21205726119861 (1205820)

Re 120577 =(1198621198891119899 minus 1198601198892119899) (1198891119899

2ℓ

2minus 119860)

21198611205726

Im 120577 =1205960 (1198621198891119899 minus 1198601198892119899)

21198611205726

(74)

Journal of Applied Mathematics 11

Since ℓ120587119886lowast

119899= 1 minus 119894(119860 minus 1198891119899

2ℓ2)1205960 and ℓ120587119887lowast

119899= minus1198941198611205960

one can get

Re ⟨119902lowast 11987611990820119902

⟩ =14[1198911119906119906 (Re 120585 + 2 Re 120591)

+1198911119906V (Re 120578 + 2 Re120594) +1198911VV (Re 120574 + 2 Re 120577)]

+(119860 minus 1198891119899

2ℓ2)

41205960[1198911119906119906 (Im 120585 + 2 Im 120591)

+1198911119906V (Im 120578 + 2 Im120594) +1198911VV (Im 120574 + 2 Im 120577)]

+119861

41205960[1198912119906119906 (Im 120585 + 2 Im 120591)

+1198912119906V (Im 120578 + 2 Im120594) +1198912VV (Im 120574 + 2 Im 120577)]

Re ⟨119902lowast 11987611990811119902

⟩ =14[1198911119906119906 (120585 + 2120591)

+1198911119906V (Re 120578 + 2 Re120594) +1198911VV (Re 120574 + 2 Re 120577)]

+(119860 minus 1198891119899

2ℓ2)

41205960[1198911119906119906 (120585 + 2120591)

+1198911119906V (Re 120578 + 2 Re120594) +1198911VV (Re 120574 + 2 Re 120577)]

+119861

41205960[1198912119906V (Im 120578 + 2 Im120594)

+1198912VV (Im 120574 + 2 Im 120577)]

(75)

In addition it follows from intℓ120587

0 cos4(119899ℓ)119909 119889119909 = 3ℓ1205878and (62) that

⟨119902lowast 119862119902119902119902

⟩ =3ℓ1205878

(119886lowast

1198991198901119899 + 119887

lowast

1198991198902119899) (76)

Therefore

Re ⟨119902lowast 119862119902119902119902

⟩ =381198612 1198911119906119906119906119861

2

minus 31198911119906119906V (119860minus1198891119899

2

ℓ2)119861+ 21198911119906VV (119860minus

11988911198992

ℓ2)

2

+1198911VVV (119860minus1198891119899

2

ℓ2)119862+(119860minus

11988911198992

ℓ2)

sdot [1198911119906119906V119861minus 21198911119906VV (119860minus1198891119899

2

ℓ2)minus1198911VVV119862]

+119861[1198912119906119906V119861minus 21198912119906VV (119860minus1198891119899

2

ℓ2)minus1198912VVV119862]

(77)

Now by (40) we have

Re 1198881 (1205820) = Re ⟨119902lowast 11987611990811119902

⟩+12Re ⟨119902lowast 119876

11990820119902⟩+

12

sdotRe ⟨119902lowast 119862119902119902119902

⟩

=18[1198911119906119906 (Re 120585 + 2 Re 120591 + 2120585 + 4120591)

+1198911119906V (Re 120578 + 2 Re120594+ 2 Re 120578 + 4 Re120594)

+1198911VV (Re 120574 + 2 Re 120577 + 2 Re 120574 + 4 Re 120577)]

+(119860 minus 1198891119899

2ℓ2)

81205960[1198911119906119906 (Im 120585 + 2 Im 120591 + 2120585 + 4120591)

+1198911119906V (Im 120578 + 2 Im120594+ 2 Re 120578 + 4 Re120594)

+1198911VV (Im 120574 + 2 Im 120577 + 2 Re 120574 + 4 Re 120577)]

+119861

81205960[1198912119906119906 (Im 120585 + 2 Im 120591)

+1198912119906V (Im 120578119868+ 2 Im120594+ 2 Im 120578 + 4 Im120594)

+1198912VV (Im 120574 + Im 120577 + 2 Im 120574 + 4 Im 120577)]

+3

161198612 11989111199061199061199061198612minus 31198911119906119906V (119860minus

11988911198992

ℓ2)119861

+ 21198911119906VV (119860minus1198891119899

2

ℓ2)

2

+1198911VVV (119860minus1198891119899

2

ℓ2)119862

+(119860minus1198891119899

2

ℓ2)

sdot [1198911119906119906V119861minus 21198911119906VV (119860minus1198891119899

2

ℓ2)minus1198911VVV119862]

+119861[1198912119906119906V119861minus 21198912119906VV (119860minus1198891119899

2

ℓ2)minus1198912VVV119862]

(78)

Thus we have the following result

Theorem 5 Assume that condition (18) holds for 119899 isin N Thenthe spatially nonhomogeneous Hopf bifurcation of system (1) at(1205820 0 0) is supercritical (resp subcritical) if

Re 1198881 (1205820)

1198601015840 (1205820) + 1198631015840 (1205820)lt 0 (119903119890119904119901 gt 0) (79)

here Re 1198881(1205820) is given by (78)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

12 Journal of Applied Mathematics

Acknowledgments

Cun-Hua Zhang was supported by the Natural ScienceFoundation of Gansu Province (1212RJZA065) Xiang-PingYan was supported by the National Natural Science Foun-dation of China (11261028) Gansu Province Natural ScienceFoundation (145RJZA216) and China Scholarship Council

References

[1] A Andronov E A Leontovich I I Gordon and A G MaierTheory of Bifurcations of Dynamical Systems on a Plane IsraelProgram for Scientific Translations Jerusalem Israel 1971

[2] J Hale Theory of Functional Differential Equations SpringerNew York NY USA 1977

[3] B D Hassard N D Kazarinoff and Y H Wan Theory andApplications of Hopf Bifurcation Cambridge University PressCambridge UK 1981

[4] H Kielhofer Bifurcation Theory An Introduction with Appli-cations to PDEs vol 156 of Applied Mathematical SciencesSpringer New York NY USA 2004

[5] YA KuznetsovElements of Applied BifurcationTheory AppliedMathematical Sciences Springer New York NY USA 1998

[6] J-H Wu Theory and Applications of Partial Functional Differ-ential Equations Springer New York NY USA 1996

[7] F Yi JWei and J Shi ldquoBifurcation and spatiotemporal patternsin a homogeneous diffusive predator-prey systemrdquo Journal ofDifferential Equations vol 246 no 5 pp 1944ndash1977 2009

[8] L-J Yang and X-W Zeng ldquoStability of singular Hopf bifurca-tionsrdquo Journal of Differential Equations vol 206 no 1 pp 30ndash54 2004

[9] J Y Jin J P Shi J J Wei and F Q Yi ldquoBifurcations of patternedsolutions in the diffusive Lengyel-Epstein system of CIMAchemical reactionsrdquo Rocky Mountain Journal of Mathematicsvol 43 no 5 pp 1637ndash1674 2013

[10] S G Ruan ldquoDiffusion-driven instability in the Gierer-Mein-hardtmodel ofmorphogenesisrdquoNatural ResourceModeling vol11 no 2 pp 131ndash142 1998

[11] F Yi J Wei and J Shi ldquoDiffusion-driven instability andbifurcation in the Lengyel-Epstein systemrdquo Nonlinear AnalysisReal World Applications vol 9 no 3 pp 1038ndash1051 2008

[12] F Q Yi J J Wei and J P Shi ldquoGlobal asymptotical behavior ofthe Lengyel-Epstein reaction-diffusion systemrdquoApplied Mathe-matics Letters vol 22 no 1 pp 52ndash55 2009

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Stochastic AnalysisInternational Journal of

Page 5: Research Article Normal Forms of Hopf Bifurcation for a ...

Journal of Applied Mathematics 5

3 Spatially Homogeneous Hopf Bifurcation

From the description in the previous section we know thatHopf bifurcation of (1) at (1205820 0 0) is spatially homogeneousif condition (18) holds when 119899 = 0 In the present section wecompute Re 1198881(1205820) in (40) in order to determine the directionof spatially homogeneous Hopf bifurcation and the stabilityof bifurcating periodic solutions of (1) at (1205820 0 0) followingthe algorithm described in this pervious section

Lemma 3 If condition (18) is satisfied when 119899 = 0 then11986720 =

11986711 = 0

Proof From (20) and (22) one can see

1198860 = 1

1198870 =1198941205960 minus 119860 (1205820)

119861 (1205820)

119886lowast

0 =1205960 + 119894119860 (1205820)

2ℓ1205871205960

119887lowast

0 = minus119894119861 (1205820)

2ℓ1205871205960

(42)

where

1205960 = radic119860 (1205820)119863 (1205820) minus 119861 (1205820) 119862 (1205820)

= radicminus1198602 (1205820) minus 119861 (1205820) 119862 (1205820)

(43)

Let all the partial derivatives of 119891119896(120582 119906 V) (119896 = 1 2) be

evaluated at (1205820 0 0) and let 1198881198960 1198891198960 and 119890

1198960 (119896 = 1 2) bedefined respectively by

1198881198960 = 119891

119896119906119906+ 2119891119896119906V1198870 +119891

119896VV11988720

1198891198960 = 119891

119896119906119906+119891119896119906V (1198870 + 1198870) +119891

119896VV10038161003816100381610038161198870

10038161003816100381610038162

1198901198960 = 119891

119896119906119906119906+119891119896119906119906V (21198870 + 1198870) +119891

119896119906VV (210038161003816100381610038161198870

10038161003816100381610038162+ 119887

20 )

+119891119896VVV

10038161003816100381610038161198870100381610038161003816100381621198870

(44)

Then from (30) and (31) we can get

119876119902119902

= (11988810

11988820)

119876119902119902

= (11988910

11988920)

119862119902119902119902

= (11989010

11989020)

(45)

Therefore

⟨119902lowast 119876119902119902⟩ = int

ℓ120587

0(119886lowast

0 11988810 + 119887lowast

0 11988820)

= ℓ120587 (119886lowast

0 11988810 + 119887lowast

0 11988820)

⟨119902lowast 119876119902119902⟩ = int

ℓ120587

0(119886lowast

0 11988810 + 119887lowast

0 11988820)

= ℓ120587 (119886lowast

0 11988810 + 119887lowast

0 11988820)

⟨119902lowast 119876119902119902⟩ = int

ℓ120587

0(119886lowast

011988910 + 119887lowast

011988920)

= ℓ120587 (119886lowast

011988910 + 119887lowast

011988920)

⟨119902lowast 119876119902119902⟩ = int

ℓ120587

0(119886lowast

0 11988910 + 119887lowast

0 11988920)

= ℓ120587 (119886lowast

0 11988910 + 119887lowast

0 11988920)

(46)

From (34) and (46) one can obtain

11986720

= (

11988810 minus ℓ120587 [(119886lowast0 + 119886lowast

0 ) 11988810 + (119887lowast0 + 119887lowast

0) 11988820]

11988820 minus ℓ120587 [(119886lowast

01198870 + 119886lowast0 1198870) 11988810 + (119887lowast

01198870 + 119887lowast0 1198870) 11988820]

)

11986711

= (

11988910 minus ℓ120587 [(119886lowast0 + 119886lowast

0 ) 11988910 + (119887lowast0 + 119887lowast

0) 11988920]

11988920 minus ℓ120587 [(119886lowast

01198870 + 119886lowast0 1198870) 11988910 + (119887lowast

01198870 + 119887lowast0 1198870) 11988920]

)

(47)

Notice from (42) that

119886lowast

0 + 119886lowast

0 = 119887lowast

01198870 + 119887lowast

0 1198870 =1ℓ120587

119887lowast

0 + 119887lowast

0 = 119886lowast

01198870 + 119886lowast

0 1198870 = 0(48)

The conclusion follows by substituting (48) into (47)

Lemma 3 and (36) imply 11990820 = 11990811 = 0 when 119899 = 0 in(18) and thus we have

11989221 = ⟨119902lowast 119862119902119902119902

⟩ = intℓ120587

0(119886lowast

0 11989010 + 119887lowast

0 11989020)

= ℓ120587 (119886lowast

0 11989010 + 119887lowast

0 11989020)

(49)

It follows from (40) that

2 Re 1198881 (1205820) = Re( 119894

1205960⟨119902lowast 119876119902119902⟩ ⟨119902lowast 119876119902119902⟩

+⟨119902lowast 119862119902119902119902

⟩)

6 Journal of Applied Mathematics

= Re[

[

119894ℓ120587 (119886lowast

0 11988810 + 119887lowast

0 11988820) ℓ120587 (119886lowast

011988910 + 119887lowast

011988920)

1205960

+ ℓ120587 (119886lowast

0 11989010 + 119887lowast

0 11989020)]

]

(50)

We represent 119860(1205820) 119861(1205820) and 119862(1205820) by 119860 119861 and119862 respectively for the simplicity of notations and underassumption (18) with 119899 = 0 substituting 1198870 in (42) into (44)yields that for 119896 = 1 2

Re 1198881198960 =

119891119896119906119906

1198612 minus 2119891119896119906V119860119861 + 119891

119896VV (21198602 + 119861119862)

1198612

Im 1198881198960 = 21205960

119891119896119906V119861 minus 119891

119896VV119860

1198612

1198891198960 =

119891119896119906119906

119860 minus 2119891119896119906V119861 minus 119891

119896VV119862

119861

Re 1198901198960 =

119891119896119906119906119906

1198612minus 3119891119896119906119906V119860119861 + 2119891

119896119906VV1198602+ 119891119896VVV119860119862

1198612

Im 1198901198960 = 1205960

119891119896119906119906V119861 minus 2119891

119896119906VV119860 minus 119891119896VVV119862

1198612

(51)

From (42) (50) and (51) one can derive

2 Re 1198881 (1205820) =C0 (1205820)

412059620119861

2 (52)

where

C0 (1205820) = [11989111199061199061198602+ (1198912119906119906 minus 21198911119906V) 119860119861minus1198911VV119860119862

minus 21198912119906V1198612minus1198912VV119861119862] (1198911119906119906119861+1198911VV119862+ 21198912119906V119861

minus 21198912VV119860)+ (1198911119906119906119860minus 21198911119906V119861minus1198911VV119862) [1198911119906119906119860119861

+ 21198911119906V119861119862minus1198911VV119860119862+11989121199061199061198612minus 21198912119906V119860119861

+1198912VV (21198602+119861119862)] minus 2 (119860

2+119861119862)

sdot [(1198911119906119906119906 +1198912119906119906V) 1198612minus 2 (1198911119906119906V +1198912119906VV) 119860119861

minus1198912VVV119861119862]

(53)

Thus we have the following result

Theorem4 Assume that condition (18) is satisfied when 119899 = 0andC0(1205820) is defined by (53) Then the spatially homogeneousHopf bifurcation of system (1) at (1205820 0 0) is supercritical (respsubcritical) if

C0 (1205820)

1198601015840 (1205820) + 1198631015840 (1205820)lt 0 (119903119890119904119901 gt 0) (54)

Moreover if each eigenvalue of 119871119895(1205820) has negative real parts

for all 119895 isin N then the above spatially homogeneous bifurcatingperiodic solutions are stable (resp unstable) when

C0 (1205820) lt 0 (119903119890119904119901 gt 0) (55)

4 Spatially NonhomogeneousHopf Bifurcation

Notice that the spatially nonhomogeneous periodic solu-tions of (1) at (1205820 0 0) from Hopf bifurcation are unstableAccordingly in this section we will calculate Re 1198881(1205820) in (40)in order to determine the direction of Hopf bifurcation ofspatially nonhomogeneous periodic solutions of system (1) at(1205820 0 0) To this end we always assume that 119899 isin N in (18)throughout this section and still represent 119860(1205820) 119861(1205820) and119862(1205820) by119860 119861 and119862 respectivelyThus 119902lowast defined in (22) hasthe form

119902lowast= (

119886lowast119899

119887lowast119899

) cos 119899

ℓ119909

= (

1205960 + 119894 (119860 minus 11988911198992ℓ2)

ℓ1205871205960

minus119894119861

ℓ1205871205960

) cos 119899

ℓ119909

(56)

where

1205960 = radic(119860 minus1198891119899

2

ℓ2)(119863 minus

11988921198992

ℓ2) minus 119861119862

= radicminus(119860 minus1198891119899

2

ℓ2)

2

minus 119861119862

(57)

Since when 119899 isin N

intℓ120587

0cos3 119899

ℓ119909 119889119909 = 0 (58)

one can obtain

⟨119902lowast 119876119902119902⟩ = ⟨119902

lowast 119876119902119902⟩ = ⟨119902

lowast 119876119902119902⟩ = ⟨119902

lowast 119876119902119902⟩

= 0(59)

Thus in order to calculate Re 1198881(1205820) it remains to compute

⟨119902lowast 11987611990811119902

⟩

⟨119902lowast 11987611990820119902

⟩

⟨119902lowast 119862119902119902119902

⟩

(60)

Journal of Applied Mathematics 7

Let all the second- and third-order partial derivatives of119891119896(120582 119906 V) (119896 = 1 2) with respect to 119906 and V be evaluated at

(1205820 0 0) and let

119888119896119899

= 119891119896119906119906

+ 2119891119896119906V119887119899 +119891

119896VV1198872119899

119889119896119899

= 119891119896119906119906

+119891119896119906V (119887119899 + 119887

119899) +119891119896VV

100381610038161003816100381611988711989910038161003816100381610038162

119890119896119899

= 119891119896119906119906119906

+119891119896119906119906V (2119887119899 + 119887

119899) +119891119896119906VV (2

100381610038161003816100381611988711989910038161003816100381610038162+ 119887

2119899)

+119891119896VVV

100381610038161003816100381611988711989910038161003816100381610038162119887119899

119896 = 1 2

(61)

Then from (30) and (31) one can observe

119876119902119902

= (1198881119899

1198882119899) cos2 119899

ℓ119909 =

12(1198881119899

1198882119899)(1+ cos 2119899

ℓ119909)

119876119902119902

= (1198891119899

1198892119899) cos2 119899

ℓ119909 =

12(1198891119899

1198892119899)(1+ cos 2119899

ℓ119909)

119862119902119902119902

= (1198901119899

1198902119899) cos3 119899

ℓ119909

(62)

In view of (34) (59) and (62) we have

11986720 = 119876119902119902

=12(1198881119899

1198882119899)(1+ cos 2119899

ℓ119909)

11986711 = 119876119902119902

=12(1198891119899

1198892119899)(1+ cos 2119899

ℓ119909)

(63)

Equalities (63) show that the calculation of [21198941205960 minus 119871(1205820)]minus1

and [119871(1205820)]minus1 will be restricted on the subspaces spanned by

eigenmodes 1 and cos(2119899ℓ)119909Let

1205721 =(1211988911198892 minus 31198892

1) 1198994

ℓ4minus3 (1198892 minus 1198891) 1198601198992

ℓ2minus 31205962

0

1205722 =6 (1198891 + 1198892) 119899

21205960ℓ2

1205723 =11988921119899

4

ℓ4+

(1198892 minus 1198891) 1198601198992

ℓ2minus 31205962

0

1205724 = minus2 (1198891 + 1198892) 119899

21205960ℓ2

(64)

Then under condition (18) one can derive

[21198941205960119868 minus 1198712119899 (1205820)]minus1

=1

1205721 + 1198941205722

sdot(

21198941205960 + 119860 +(31198892 minus 1198891) 119899

2

ℓ2119861

119862 21198941205960 minus 119860 +41198891119899

2

ℓ2

)

[21198941205960119868 minus 1198710 (1205820)]minus1

=1

1205723 + 1198941205723

sdot(21198941205960 + 119860 minus

(1198891 + 1198892) 1198992

ℓ2119861

119862 21198941205960 minus 119860

)

(65)

From (36) and (61) we have

11990820 = [21198941205960119868 minus 1198712119899 (1205820)]

minus1

2cos 2119899

ℓ119909

+[21198941205960119868 minus 1198710 (1205820)]

minus1

2(

1198881119899

1198882119899) =

12 (1205721 + 1198941205722)

sdot(

[21198941205960 + 119860 +(31198892 minus 1198891) 119899

2

ℓ2] 1198881119899 + 1198611198882119899

1198621198881119899 + (21198941205960 minus 119860 +41198891119899

2

ℓ2) 1198882119899

)

sdot cos 2119899ℓ

119909 +1

2 (1205723 + 1198941205724)

sdot([21198941205960 + 119860 minus

(1198891 + 1198892) 1198992

ℓ2] 1198881119899 + 1198611198882119899

1198621198881119899 + (21198941205960 minus 119860) 1198882119899

)

(66)

Similarly we can get

11990811 =121205725

(

[119860 +(31198892 minus 1198891) 119899

2

ℓ2]1198891119899 + 1198611198892119899

1198621198891119899 + (41198891119899

2

ℓ2minus 119860)1198892119899

)

sdot cos 2119899ℓ

119909 +121205726

sdot([119860 minus

(1198891 + 1198892) 1198992

ℓ2]1198891119899 + 1198611198892119899

1198621198891119899 minus 1198601198892119899

)

(67)

where

1205725 =(1211988911198892 minus 31198892

1) 1198994

ℓ4minus3 (1198892 minus 1198891) 119860119899

2

ℓ2+120596

20

1205726 =11988921119899

4

ℓ4+

(1198892 minus 1198891) 1198601198992

ℓ2+120596

20

(68)

8 Journal of Applied Mathematics

From (30) we have

11987611990820119902

= (1198911119906119906120585 + 1198911119906V120578 + 1198911VV120574

1198912119906119906120585 + 1198912119906V120578 + 1198912VV120574) cos 119899

ℓ119909 cos 2119899

ℓ119909

+(1198911119906119906120591 + 1198911119906V120594 + 1198911VV120577

1198912119906119906120591 + 1198912119906V120594 + 1198912VV120577) cos 119899

ℓ119909

11987611990811119902

= (1198911119906119906120585 + 1198911119906V120578 + 1198911VV120574

1198912119906119906120585 + 1198912119906V120578 + 1198912VV120574) cos 119899

ℓ119909 cos 2119899

ℓ119909

+(1198911119906119906120591 + 1198911119906V120594 + 1198911VV120577

1198912119906119906120591 + 1198912119906V120594 + 1198912VV120577) cos 119899

ℓ119909

(69)

with

120585 =(21198941205960 + 119860 + (31198892 minus 1198891) 119899

2ℓ

2) 1198881119899 + 1198611198882119899

2 (1205721 + 1198941205722)

120578 =(21198941205960 + 119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860 minus 1198941205960) + 119861119862

2119861 (1205721 + 1198941205722)

sdot 1198881119899 +(51198891119899

2ℓ2 minus 2119860 + 1198941205960)

2 (1205721 + 1198941205722)1198882119899

120574 =[1198621198881119899 + (21198941205960 minus 119860 + 41198891119899

2ℓ2) 1198882119899] (11988911198992ℓ2 minus 119860 minus 1198941205960)

2119861 (1205721 + 1198941205722)

120591 =(21198941205960 + 119860 minus (1198891 + 1198892) 119899

2ℓ2) 1198881119899 + 1198611198882119899

2 (1205723 + 1198941205724)

120594 =(21198941205960 + 119860 minus (1198891 + 1198892) 119899

2ℓ2) (11988911198992ℓ2 minus 119860 minus 1198941205960) + 119861119862

2119861 (1205723 + 1198941205724)

sdot 1198881119899 +(1198891119899

2ℓ2 minus 2119860 + 1198941205960)

2 (1205723 + 1198941205724)1198882119899

120577 =[1198621198881119899 + (21198941205960 minus 119860) 1198882119899] (1198891119899

2ℓ2 minus 119860 minus 1198941205960)

2119861 (1205723 + 1198941205724)

120585 =(119860 + (31198892 minus 1198891) 119899

2ℓ2) 1198891119899 + 1198611198892119899

21205725

120578 =(119860 + (31198892 minus 1198891) 119899

2ℓ2) (1198941205960 minus 119860 + 11988911198992ℓ2) + 119861119862

211986112057251198891119899

+(1198941205960 minus 2119860 + 51198891119899

2ℓ2)

212057251198892119899

120574 =(1198941205960 minus 119860 + 1198891119899

2ℓ2) [1198621198891119899 + (411988911198992ℓ2 minus 119860) 1198892119899]

21198611205725

120591 =(119860 minus (1198891 + 1198892) 119899

2ℓ2) 1198891119899 + 1198611198892119899

21205726

120594 =(119860 minus (1198891 + 1198892) 119899

2ℓ2) (1198941205960 minus 119860 + 11988911198992ℓ2) + 119861119862

211986112057261198891119899

+(1198941205960 minus 2119860 + 1198891119899

2ℓ2)

212057261198892119899

120577 =(1198941205960 minus 119860 + 1198891119899

2ℓ2) (1198621198891119899 minus 1198601198892119899)

21198611205726

(70)

Notice that for 119899 isin N

intℓ120587

0cos2 119899

ℓ119909 119889119909 =

ℓ120587

2

intℓ120587

0cos 2119899

ℓ119909cos2 119899

ℓ119909 119889119909 =

ℓ120587

4

(71)

It follows from (69) that

⟨119902lowast 11987611990820119902

⟩ =ℓ120587

4[119886lowast

119899(1198911119906119906120585 +1198911119906V120578 +1198911VV120574)

+ 119887lowast

119899(1198912119906119906120585 +1198912119906V120578 +1198912VV120574)]

+ℓ120587

2[119886lowast

119899(1198911119906119906120591 +1198911119906V120594+1198911VV120577)

+ 119887lowast

119899(1198912119906119906120591 +1198912119906V120594+1198912VV120577)]

⟨119902lowast 11987611990811119902

⟩ =ℓ120587

4[119886lowast

119899(1198911119906119906120585 +1198911119906V120578 +1198911VV120574)

+ 119887lowast

119899(1198912119906119906120585 +1198912119906V120578 +1198912VV120574)]

+ℓ120587

2[119886lowast

119899(1198911119906119906120591 +1198911119906V120594+1198911VV120577)

+ 119887lowast

119899(1198912119906119906120591 +1198912119906V120594+1198912VV120577)]

(72)

Substituting 119887119899= (1198941205960minus119860(1205820)+1198891119899

2ℓ2)119861(1205820) into (61) gives

Re 119888119896119899

=119891119896119906119906

1198612 minus 2119891119896119906V (119860 minus 1198891119899

2ℓ2) 119861 + 119891119896VV [2 (119860 minus 1198891119899

2ℓ2)2+ 119861119862]

1198612

Im 119888119896119899

=21205960 [119891119896119906V119861 minus 119891

119896VV (119860 minus 11988911198992ℓ2)]

1198612

Journal of Applied Mathematics 9

119889119896119899

=119891119896119906119906

(119860 minus 11988911198992ℓ2) minus 2119891

119896119906V119861 minus 119891119896VV119862

119861

Re 119890119896119899

=119891119896119906119906119906

1198612 minus 3119891119896119906119906V (119860 minus 1198891119899

2ℓ2) 119861 + 2119891119896119906VV (119860 minus 1198891119899

2ℓ2)2+ 119891119896VVV (119860 minus 1198891119899

2ℓ2) 119862

1198612

Im 119890119896119899

=1205960 [119891119896119906119906V119861 minus 2119891

119896119906VV (119860 minus 11988911198992ℓ2) minus 119891

119896VVV119862]

1198612

119896 = 1 2(73)

Then from (70) and (73) we have

Re 120585 =[(119860 + (31198892 minus 1198891) 119899

2ℓ2)Re 1198881119899 + 119861Re 1198882119899 minus 21205960 Im 1198881119899] 1205721 + [(119860 + (31198892 minus 1198891) 119899

2ℓ2) Im 1198881119899 + 119861 Im 1198882119899 + 21205960 Re 1198881119899] 1205722

2 (12057221 + 1205722

2)

Im 120585 =[(119860 + (31198892 minus 1198891) 119899

2ℓ2) Im 1198881119899 + 119861 Im 1198882119899 + 21205960 Re 1198881119899] 1205721 minus [(119860 + (31198892 minus 1198891) 1198992ℓ2)Re 1198881119899 + 119861Re 1198882119899 minus 21205960 Im 1198881119899] 1205722

2 (12057221 + 1205722

2)

Re 120578

=[(119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0]Re 1198881119899 minus 31205960 ((1198891 minus 1198892) 1198992ℓ2 minus 119860) Im 1198881119899 + 119861 (51198891119899

2ℓ2 minus 2119860)Re 1198882119899 minus 1198611205960 Im 1198882119899

2119861 (12057221 + 1205722

2)

sdot 1205721

+[(119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0] Im 1198881119899 + 31205960 ((1198891 minus 1198892) 1198992ℓ2 minus 119860)Re 1198881119899 + 119861 (51198891119899

2ℓ2 minus 2119860) Im 1198882119899 + 1198611205960 Re 11988821198992119861 (1205722

1 + 12057222)

sdot 1205722

Im 120578

=[(119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0] Im 1198881119899 + 31205960 ((1198891 minus 1198892) 1198992ℓ2 minus 119860)Re 1198881119899 + 119861 (51198891119899

2ℓ2 minus 2119860) Im 1198882119899 + 1198611205960 Re 11988821198992119861 (1205722

1 + 12057222)

sdot 1205721

minus[(119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0]Re 1198881119899 minus 31205960 ((1198891 minus 1198892) 1198992ℓ2 minus 119860) Im 1198881119899 + 119861 (51198891119899

2ℓ2 minus 2119860)Re 1198882119899 minus 1198611205960 Im 1198882119899

2119861 (12057221 + 1205722

2)

sdot 1205722

Re 120574 =(1198891119899

2ℓ2 minus 119860) [119862Re 1198881119899 minus (119860 minus 411988911198992ℓ2)Re 1198882119899 minus 21205960 Im 1198882119899] + 1205960 [119862 Im 1198881119899 minus (119860 minus 41198891119899

2ℓ2) Im 1198882119899 + 21205960 Re 1198882119899]2119861 (1205722

1 + 12057222)

1205721

+(1198891119899

2ℓ2 minus 119860) [119862 Im 1198881119899 minus (119860 minus 411988911198992ℓ2) Im 1198882119899 + 21205960 Re 1198882119899] minus 1205960 [119862Re 1198881119899 minus (119860 minus 41198891119899

2ℓ2)Re 1198882119899 minus 21205960 Im 1198882119899]

2119861 (12057221 + 1205722

2)1205722

Im 120574 =(1198891119899

2ℓ2 minus 119860) [119862 Im 1198881119899 minus (119860 minus 411988911198992ℓ2) Im 1198882119899 + 21205960 Re 1198882119899] minus 1205960 [119862Re 1198881119899 minus (119860 minus 41198891119899

2ℓ2)Re 1198882119899 minus 21205960 Im 1198882119899]

2119861 (12057221 + 1205722

2)1205721

10 Journal of Applied Mathematics

minus(1198891119899

2ℓ2 minus 119860) [119862Re 1198881119899 minus (119860 minus 411988911198992ℓ2)Re 1198882119899 minus 21205960 Im 1198882119899] + 1205960 [119862 Im 1198881119899 minus (119860 minus 41198891119899

2ℓ2) Im 1198882119899 + 21205960 Re 1198882119899]2119861 (1205722

1 + 12057222)

1205722

Re 120591 =[(119860 + (1198891 + 1198892) 119899

2ℓ

2)Re 1198881119899 + 119861Re 1198882119899 minus 21205960 Im 1198881119899] 1205723 + [(119860 + (1198891 + 1198892) 119899

2ℓ

2) Im 1198881119899 + 119861 Im 1198882119899 + 21205960 Re 1198881119899] 1205724

2 (12057223 + 1205722

4)

Im 120591 =[(119860 + (1198891 + 1198892) 119899

2ℓ2) Im 1198881119899 + 119861 Im 1198882119899 + 21205960 Re 1198881119899] 1205723 minus [(119860 + (1198891 + 1198892) 1198992ℓ2)Re 1198881119899 + 119861Re 1198882119899 minus 21205960 Im 1198881119899] 1205724

2 (12057223 + 1205722

4)

Re120594

=[(119860 minus (1198891 + 1198892) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0]Re 1198881119899 minus 1205960 ((31198891 + 1198892) 1198992ℓ2 minus 3119860) Im 1198881119899 + 119861 (1198891119899

2ℓ2 minus 2119860)Re 1198882119899 minus 1198611205960 Im 1198882119899

2119861 (12057223 + 1205722

4)

sdot 1205723

+[(119860 minus (1198891 + 1198892) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0] Im 1198881119899 + 1205960 ((31198891 + 1198892) 1198992ℓ2 minus 3119860)Re 1198881119899 + 119861 (1198891119899

2ℓ2 minus 2119860) Im 1198882119899 + 1198611205960 Re 11988821198992119861 (1205722

3 + 12057224)

sdot 1205724

Im120594

=[(119860 minus (1198891 + 1198892) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0] Im 1198881119899 + 1205960 ((31198891 + 1198892) 1198992ℓ2 minus 3119860)Re 1198881119899 + 119861 (1198891119899

2ℓ2 minus 2119860) Im 1198882119899 + 1198611205960 Re 11988821198992119861 (1205722

3 + 12057224)

sdot 1205723

minus[(119860 minus (1198891 + 1198892) 119899

2ℓ

2) (1198891119899

2ℓ

2minus 119860) + 119861119862 + 21205962

0]Re 1198881119899 minus 1205960 ((31198891 + 1198892) 1198992ℓ

2minus 3119860) Im 1198881119899 + 119861 (1198891119899

2ℓ

2minus 2119860)Re 1198882119899 minus 1198611205960 Im 1198882119899

2119861 (12057223 + 1205722

4)

sdot 1205724

Re 120577 =(1198891119899

2ℓ

2minus 119860) (119862Re 1198881119899 minus 119860Re 1198882119899 minus 21205960 Im 1198882119899) + 1205960 (119862 Im 1198881119899 minus 119860 Im 1198882119899 + 21205960 Re 1198882119899)

2119861 (12057223 + 1205722

4)1205723

+(1198891119899

2ℓ2 minus 119860) (119862 Im 1198881119899 minus 119860 Im 1198882119899 + 21205960 Re 1198882119899) minus 1205960 (119862Re 1198881119899 minus 119860Re 1198882119899 minus 21205960 Im 1198882119899)

2119861 (12057223 + 1205722

4)1205724

Im 120577 =(1198891119899

2ℓ2 minus 119860) (119862 Im 1198881119899 minus 119860 Im 1198882119899 + 21205960 Re 1198882119899) minus 1205960 (119862Re 1198881119899 minus 119860Re 1198882119899 minus 21205960 Im 1198882119899)

2119861 (12057223 + 1205722

4)1205723

minus(1198891119899

2ℓ2 minus 119860) (119862Re 1198881119899 minus 119860Re 1198882119899 minus 21205960 Im 1198882119899) + 1205960 (119862 Im 1198881119899 minus 119860 Im 1198882119899 + 21205960 Re 1198882119899)2119861 (1205722

3 + 12057224)

1205724

Re 120578 =[(119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862] 1198891119899 + 119861 (51198891119899

2ℓ2 minus 2119860)1198892119899

21198611205725

Re 120578 =1205960 [(119860 + (31198892 minus 1198891) 119899

2ℓ2) 1198891119899 + 1198611198892119899]

21205725

Re 120574 =(1198891119899

2ℓ2 minus 119860) [1198621198891119899 + (411988911198992ℓ2 minus 119860) 1198892119899]

21198611205725

Im 120574 =1205960 [1198621198891119899 + (41198891119899

2ℓ2 minus 119860) 1198892119899]

21198611205725

Re120594 =[(119860 minus (1198891 + 1198892) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862] 1198891119899 + 119861 (1198891119899

2ℓ2 minus 2119860)1198892119899

21198611205726

Im120594 =1205960 [(119860 minus (1198891 + 1198892) 119899

2ℓ2) 1198891119899 + 1198611198892119899]

21205726119861 (1205820)

Re 120577 =(1198621198891119899 minus 1198601198892119899) (1198891119899

2ℓ

2minus 119860)

21198611205726

Im 120577 =1205960 (1198621198891119899 minus 1198601198892119899)

21198611205726

(74)

Journal of Applied Mathematics 11

Since ℓ120587119886lowast

119899= 1 minus 119894(119860 minus 1198891119899

2ℓ2)1205960 and ℓ120587119887lowast

119899= minus1198941198611205960

one can get

Re ⟨119902lowast 11987611990820119902

⟩ =14[1198911119906119906 (Re 120585 + 2 Re 120591)

+1198911119906V (Re 120578 + 2 Re120594) +1198911VV (Re 120574 + 2 Re 120577)]

+(119860 minus 1198891119899

2ℓ2)

41205960[1198911119906119906 (Im 120585 + 2 Im 120591)

+1198911119906V (Im 120578 + 2 Im120594) +1198911VV (Im 120574 + 2 Im 120577)]

+119861

41205960[1198912119906119906 (Im 120585 + 2 Im 120591)

+1198912119906V (Im 120578 + 2 Im120594) +1198912VV (Im 120574 + 2 Im 120577)]

Re ⟨119902lowast 11987611990811119902

⟩ =14[1198911119906119906 (120585 + 2120591)

+1198911119906V (Re 120578 + 2 Re120594) +1198911VV (Re 120574 + 2 Re 120577)]

+(119860 minus 1198891119899

2ℓ2)

41205960[1198911119906119906 (120585 + 2120591)

+1198911119906V (Re 120578 + 2 Re120594) +1198911VV (Re 120574 + 2 Re 120577)]

+119861

41205960[1198912119906V (Im 120578 + 2 Im120594)

+1198912VV (Im 120574 + 2 Im 120577)]

(75)

In addition it follows from intℓ120587

0 cos4(119899ℓ)119909 119889119909 = 3ℓ1205878and (62) that

⟨119902lowast 119862119902119902119902

⟩ =3ℓ1205878

(119886lowast

1198991198901119899 + 119887

lowast

1198991198902119899) (76)

Therefore

Re ⟨119902lowast 119862119902119902119902

⟩ =381198612 1198911119906119906119906119861

2

minus 31198911119906119906V (119860minus1198891119899

2

ℓ2)119861+ 21198911119906VV (119860minus

11988911198992

ℓ2)

2

+1198911VVV (119860minus1198891119899

2

ℓ2)119862+(119860minus

11988911198992

ℓ2)

sdot [1198911119906119906V119861minus 21198911119906VV (119860minus1198891119899

2

ℓ2)minus1198911VVV119862]

+119861[1198912119906119906V119861minus 21198912119906VV (119860minus1198891119899

2

ℓ2)minus1198912VVV119862]

(77)

Now by (40) we have

Re 1198881 (1205820) = Re ⟨119902lowast 11987611990811119902

⟩+12Re ⟨119902lowast 119876

11990820119902⟩+

12

sdotRe ⟨119902lowast 119862119902119902119902

⟩

=18[1198911119906119906 (Re 120585 + 2 Re 120591 + 2120585 + 4120591)

+1198911119906V (Re 120578 + 2 Re120594+ 2 Re 120578 + 4 Re120594)

+1198911VV (Re 120574 + 2 Re 120577 + 2 Re 120574 + 4 Re 120577)]

+(119860 minus 1198891119899

2ℓ2)

81205960[1198911119906119906 (Im 120585 + 2 Im 120591 + 2120585 + 4120591)

+1198911119906V (Im 120578 + 2 Im120594+ 2 Re 120578 + 4 Re120594)

+1198911VV (Im 120574 + 2 Im 120577 + 2 Re 120574 + 4 Re 120577)]

+119861

81205960[1198912119906119906 (Im 120585 + 2 Im 120591)

+1198912119906V (Im 120578119868+ 2 Im120594+ 2 Im 120578 + 4 Im120594)

+1198912VV (Im 120574 + Im 120577 + 2 Im 120574 + 4 Im 120577)]

+3

161198612 11989111199061199061199061198612minus 31198911119906119906V (119860minus

11988911198992

ℓ2)119861

+ 21198911119906VV (119860minus1198891119899

2

ℓ2)

2

+1198911VVV (119860minus1198891119899

2

ℓ2)119862

+(119860minus1198891119899

2

ℓ2)

sdot [1198911119906119906V119861minus 21198911119906VV (119860minus1198891119899

2

ℓ2)minus1198911VVV119862]

+119861[1198912119906119906V119861minus 21198912119906VV (119860minus1198891119899

2

ℓ2)minus1198912VVV119862]

(78)

Thus we have the following result

Theorem 5 Assume that condition (18) holds for 119899 isin N Thenthe spatially nonhomogeneous Hopf bifurcation of system (1) at(1205820 0 0) is supercritical (resp subcritical) if

Re 1198881 (1205820)

1198601015840 (1205820) + 1198631015840 (1205820)lt 0 (119903119890119904119901 gt 0) (79)

here Re 1198881(1205820) is given by (78)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

12 Journal of Applied Mathematics

Acknowledgments

Cun-Hua Zhang was supported by the Natural ScienceFoundation of Gansu Province (1212RJZA065) Xiang-PingYan was supported by the National Natural Science Foun-dation of China (11261028) Gansu Province Natural ScienceFoundation (145RJZA216) and China Scholarship Council

References

[1] A Andronov E A Leontovich I I Gordon and A G MaierTheory of Bifurcations of Dynamical Systems on a Plane IsraelProgram for Scientific Translations Jerusalem Israel 1971

[2] J Hale Theory of Functional Differential Equations SpringerNew York NY USA 1977

[3] B D Hassard N D Kazarinoff and Y H Wan Theory andApplications of Hopf Bifurcation Cambridge University PressCambridge UK 1981

[4] H Kielhofer Bifurcation Theory An Introduction with Appli-cations to PDEs vol 156 of Applied Mathematical SciencesSpringer New York NY USA 2004

[5] YA KuznetsovElements of Applied BifurcationTheory AppliedMathematical Sciences Springer New York NY USA 1998

[6] J-H Wu Theory and Applications of Partial Functional Differ-ential Equations Springer New York NY USA 1996

[7] F Yi JWei and J Shi ldquoBifurcation and spatiotemporal patternsin a homogeneous diffusive predator-prey systemrdquo Journal ofDifferential Equations vol 246 no 5 pp 1944ndash1977 2009

[8] L-J Yang and X-W Zeng ldquoStability of singular Hopf bifurca-tionsrdquo Journal of Differential Equations vol 206 no 1 pp 30ndash54 2004

[9] J Y Jin J P Shi J J Wei and F Q Yi ldquoBifurcations of patternedsolutions in the diffusive Lengyel-Epstein system of CIMAchemical reactionsrdquo Rocky Mountain Journal of Mathematicsvol 43 no 5 pp 1637ndash1674 2013

[10] S G Ruan ldquoDiffusion-driven instability in the Gierer-Mein-hardtmodel ofmorphogenesisrdquoNatural ResourceModeling vol11 no 2 pp 131ndash142 1998

[11] F Yi J Wei and J Shi ldquoDiffusion-driven instability andbifurcation in the Lengyel-Epstein systemrdquo Nonlinear AnalysisReal World Applications vol 9 no 3 pp 1038ndash1051 2008

[12] F Q Yi J J Wei and J P Shi ldquoGlobal asymptotical behavior ofthe Lengyel-Epstein reaction-diffusion systemrdquoApplied Mathe-matics Letters vol 22 no 1 pp 52ndash55 2009

Submit your manuscripts athttpwwwhindawicom

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 6: Research Article Normal Forms of Hopf Bifurcation for a ...

6 Journal of Applied Mathematics

= Re[

[

119894ℓ120587 (119886lowast

0 11988810 + 119887lowast

0 11988820) ℓ120587 (119886lowast

011988910 + 119887lowast

011988920)

1205960

+ ℓ120587 (119886lowast

0 11989010 + 119887lowast

0 11989020)]

]

(50)

We represent 119860(1205820) 119861(1205820) and 119862(1205820) by 119860 119861 and119862 respectively for the simplicity of notations and underassumption (18) with 119899 = 0 substituting 1198870 in (42) into (44)yields that for 119896 = 1 2

Re 1198881198960 =

119891119896119906119906

1198612 minus 2119891119896119906V119860119861 + 119891

119896VV (21198602 + 119861119862)

1198612

Im 1198881198960 = 21205960

119891119896119906V119861 minus 119891

119896VV119860

1198612

1198891198960 =

119891119896119906119906

119860 minus 2119891119896119906V119861 minus 119891

119896VV119862

119861

Re 1198901198960 =

119891119896119906119906119906

1198612minus 3119891119896119906119906V119860119861 + 2119891

119896119906VV1198602+ 119891119896VVV119860119862

1198612

Im 1198901198960 = 1205960

119891119896119906119906V119861 minus 2119891

119896119906VV119860 minus 119891119896VVV119862

1198612

(51)

From (42) (50) and (51) one can derive

2 Re 1198881 (1205820) =C0 (1205820)

412059620119861

2 (52)

where

C0 (1205820) = [11989111199061199061198602+ (1198912119906119906 minus 21198911119906V) 119860119861minus1198911VV119860119862

minus 21198912119906V1198612minus1198912VV119861119862] (1198911119906119906119861+1198911VV119862+ 21198912119906V119861

minus 21198912VV119860)+ (1198911119906119906119860minus 21198911119906V119861minus1198911VV119862) [1198911119906119906119860119861

+ 21198911119906V119861119862minus1198911VV119860119862+11989121199061199061198612minus 21198912119906V119860119861

+1198912VV (21198602+119861119862)] minus 2 (119860

2+119861119862)

sdot [(1198911119906119906119906 +1198912119906119906V) 1198612minus 2 (1198911119906119906V +1198912119906VV) 119860119861

minus1198912VVV119861119862]

(53)

Thus we have the following result

Theorem4 Assume that condition (18) is satisfied when 119899 = 0andC0(1205820) is defined by (53) Then the spatially homogeneousHopf bifurcation of system (1) at (1205820 0 0) is supercritical (respsubcritical) if

C0 (1205820)

1198601015840 (1205820) + 1198631015840 (1205820)lt 0 (119903119890119904119901 gt 0) (54)

Moreover if each eigenvalue of 119871119895(1205820) has negative real parts

for all 119895 isin N then the above spatially homogeneous bifurcatingperiodic solutions are stable (resp unstable) when

C0 (1205820) lt 0 (119903119890119904119901 gt 0) (55)

4 Spatially NonhomogeneousHopf Bifurcation

Notice that the spatially nonhomogeneous periodic solu-tions of (1) at (1205820 0 0) from Hopf bifurcation are unstableAccordingly in this section we will calculate Re 1198881(1205820) in (40)in order to determine the direction of Hopf bifurcation ofspatially nonhomogeneous periodic solutions of system (1) at(1205820 0 0) To this end we always assume that 119899 isin N in (18)throughout this section and still represent 119860(1205820) 119861(1205820) and119862(1205820) by119860 119861 and119862 respectivelyThus 119902lowast defined in (22) hasthe form

119902lowast= (

119886lowast119899

119887lowast119899

) cos 119899

ℓ119909

= (

1205960 + 119894 (119860 minus 11988911198992ℓ2)

ℓ1205871205960

minus119894119861

ℓ1205871205960

) cos 119899

ℓ119909

(56)

where

1205960 = radic(119860 minus1198891119899

2

ℓ2)(119863 minus

11988921198992

ℓ2) minus 119861119862

= radicminus(119860 minus1198891119899

2

ℓ2)

2

minus 119861119862

(57)

Since when 119899 isin N

intℓ120587

0cos3 119899

ℓ119909 119889119909 = 0 (58)

one can obtain

⟨119902lowast 119876119902119902⟩ = ⟨119902

lowast 119876119902119902⟩ = ⟨119902

lowast 119876119902119902⟩ = ⟨119902

lowast 119876119902119902⟩

= 0(59)

Thus in order to calculate Re 1198881(1205820) it remains to compute

⟨119902lowast 11987611990811119902

⟩

⟨119902lowast 11987611990820119902

⟩

⟨119902lowast 119862119902119902119902

⟩

(60)

Journal of Applied Mathematics 7

Let all the second- and third-order partial derivatives of119891119896(120582 119906 V) (119896 = 1 2) with respect to 119906 and V be evaluated at

(1205820 0 0) and let

119888119896119899

= 119891119896119906119906

+ 2119891119896119906V119887119899 +119891

119896VV1198872119899

119889119896119899

= 119891119896119906119906

+119891119896119906V (119887119899 + 119887

119899) +119891119896VV

100381610038161003816100381611988711989910038161003816100381610038162

119890119896119899

= 119891119896119906119906119906

+119891119896119906119906V (2119887119899 + 119887

119899) +119891119896119906VV (2

100381610038161003816100381611988711989910038161003816100381610038162+ 119887

2119899)

+119891119896VVV

100381610038161003816100381611988711989910038161003816100381610038162119887119899

119896 = 1 2

(61)

Then from (30) and (31) one can observe

119876119902119902

= (1198881119899

1198882119899) cos2 119899

ℓ119909 =

12(1198881119899

1198882119899)(1+ cos 2119899

ℓ119909)

119876119902119902

= (1198891119899

1198892119899) cos2 119899

ℓ119909 =

12(1198891119899

1198892119899)(1+ cos 2119899

ℓ119909)

119862119902119902119902

= (1198901119899

1198902119899) cos3 119899

ℓ119909

(62)

In view of (34) (59) and (62) we have

11986720 = 119876119902119902

=12(1198881119899

1198882119899)(1+ cos 2119899

ℓ119909)

11986711 = 119876119902119902

=12(1198891119899

1198892119899)(1+ cos 2119899

ℓ119909)

(63)

Equalities (63) show that the calculation of [21198941205960 minus 119871(1205820)]minus1

and [119871(1205820)]minus1 will be restricted on the subspaces spanned by

eigenmodes 1 and cos(2119899ℓ)119909Let

1205721 =(1211988911198892 minus 31198892

1) 1198994

ℓ4minus3 (1198892 minus 1198891) 1198601198992

ℓ2minus 31205962

0

1205722 =6 (1198891 + 1198892) 119899

21205960ℓ2

1205723 =11988921119899

4

ℓ4+

(1198892 minus 1198891) 1198601198992

ℓ2minus 31205962

0

1205724 = minus2 (1198891 + 1198892) 119899

21205960ℓ2

(64)

Then under condition (18) one can derive

[21198941205960119868 minus 1198712119899 (1205820)]minus1

=1

1205721 + 1198941205722

sdot(

21198941205960 + 119860 +(31198892 minus 1198891) 119899

2

ℓ2119861

119862 21198941205960 minus 119860 +41198891119899

2

ℓ2

)

[21198941205960119868 minus 1198710 (1205820)]minus1

=1

1205723 + 1198941205723

sdot(21198941205960 + 119860 minus

(1198891 + 1198892) 1198992

ℓ2119861

119862 21198941205960 minus 119860

)

(65)

From (36) and (61) we have

11990820 = [21198941205960119868 minus 1198712119899 (1205820)]

minus1

2cos 2119899

ℓ119909

+[21198941205960119868 minus 1198710 (1205820)]

minus1

2(

1198881119899

1198882119899) =

12 (1205721 + 1198941205722)

sdot(

[21198941205960 + 119860 +(31198892 minus 1198891) 119899

2

ℓ2] 1198881119899 + 1198611198882119899

1198621198881119899 + (21198941205960 minus 119860 +41198891119899

2

ℓ2) 1198882119899

)

sdot cos 2119899ℓ

119909 +1

2 (1205723 + 1198941205724)

sdot([21198941205960 + 119860 minus

(1198891 + 1198892) 1198992

ℓ2] 1198881119899 + 1198611198882119899

1198621198881119899 + (21198941205960 minus 119860) 1198882119899

)

(66)

Similarly we can get

11990811 =121205725

(

[119860 +(31198892 minus 1198891) 119899

2

ℓ2]1198891119899 + 1198611198892119899

1198621198891119899 + (41198891119899

2

ℓ2minus 119860)1198892119899

)

sdot cos 2119899ℓ

119909 +121205726

sdot([119860 minus

(1198891 + 1198892) 1198992

ℓ2]1198891119899 + 1198611198892119899

1198621198891119899 minus 1198601198892119899

)

(67)

where

1205725 =(1211988911198892 minus 31198892

1) 1198994

ℓ4minus3 (1198892 minus 1198891) 119860119899

2

ℓ2+120596

20

1205726 =11988921119899

4

ℓ4+

(1198892 minus 1198891) 1198601198992

ℓ2+120596

20

(68)

8 Journal of Applied Mathematics

From (30) we have

11987611990820119902

= (1198911119906119906120585 + 1198911119906V120578 + 1198911VV120574

1198912119906119906120585 + 1198912119906V120578 + 1198912VV120574) cos 119899

ℓ119909 cos 2119899

ℓ119909

+(1198911119906119906120591 + 1198911119906V120594 + 1198911VV120577

1198912119906119906120591 + 1198912119906V120594 + 1198912VV120577) cos 119899

ℓ119909

11987611990811119902

= (1198911119906119906120585 + 1198911119906V120578 + 1198911VV120574

1198912119906119906120585 + 1198912119906V120578 + 1198912VV120574) cos 119899

ℓ119909 cos 2119899

ℓ119909

+(1198911119906119906120591 + 1198911119906V120594 + 1198911VV120577

1198912119906119906120591 + 1198912119906V120594 + 1198912VV120577) cos 119899

ℓ119909

(69)

with

120585 =(21198941205960 + 119860 + (31198892 minus 1198891) 119899

2ℓ

2) 1198881119899 + 1198611198882119899

2 (1205721 + 1198941205722)

120578 =(21198941205960 + 119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860 minus 1198941205960) + 119861119862

2119861 (1205721 + 1198941205722)

sdot 1198881119899 +(51198891119899

2ℓ2 minus 2119860 + 1198941205960)

2 (1205721 + 1198941205722)1198882119899

120574 =[1198621198881119899 + (21198941205960 minus 119860 + 41198891119899

2ℓ2) 1198882119899] (11988911198992ℓ2 minus 119860 minus 1198941205960)

2119861 (1205721 + 1198941205722)

120591 =(21198941205960 + 119860 minus (1198891 + 1198892) 119899

2ℓ2) 1198881119899 + 1198611198882119899

2 (1205723 + 1198941205724)

120594 =(21198941205960 + 119860 minus (1198891 + 1198892) 119899

2ℓ2) (11988911198992ℓ2 minus 119860 minus 1198941205960) + 119861119862

2119861 (1205723 + 1198941205724)

sdot 1198881119899 +(1198891119899

2ℓ2 minus 2119860 + 1198941205960)

2 (1205723 + 1198941205724)1198882119899

120577 =[1198621198881119899 + (21198941205960 minus 119860) 1198882119899] (1198891119899

2ℓ2 minus 119860 minus 1198941205960)

2119861 (1205723 + 1198941205724)

120585 =(119860 + (31198892 minus 1198891) 119899

2ℓ2) 1198891119899 + 1198611198892119899

21205725

120578 =(119860 + (31198892 minus 1198891) 119899

2ℓ2) (1198941205960 minus 119860 + 11988911198992ℓ2) + 119861119862

211986112057251198891119899

+(1198941205960 minus 2119860 + 51198891119899

2ℓ2)

212057251198892119899

120574 =(1198941205960 minus 119860 + 1198891119899

2ℓ2) [1198621198891119899 + (411988911198992ℓ2 minus 119860) 1198892119899]

21198611205725

120591 =(119860 minus (1198891 + 1198892) 119899

2ℓ2) 1198891119899 + 1198611198892119899

21205726

120594 =(119860 minus (1198891 + 1198892) 119899

2ℓ2) (1198941205960 minus 119860 + 11988911198992ℓ2) + 119861119862

211986112057261198891119899

+(1198941205960 minus 2119860 + 1198891119899

2ℓ2)

212057261198892119899

120577 =(1198941205960 minus 119860 + 1198891119899

2ℓ2) (1198621198891119899 minus 1198601198892119899)

21198611205726

(70)

Notice that for 119899 isin N

intℓ120587

0cos2 119899

ℓ119909 119889119909 =

ℓ120587

2

intℓ120587

0cos 2119899

ℓ119909cos2 119899

ℓ119909 119889119909 =

ℓ120587

4

(71)

It follows from (69) that

⟨119902lowast 11987611990820119902

⟩ =ℓ120587

4[119886lowast

119899(1198911119906119906120585 +1198911119906V120578 +1198911VV120574)

+ 119887lowast

119899(1198912119906119906120585 +1198912119906V120578 +1198912VV120574)]

+ℓ120587

2[119886lowast

119899(1198911119906119906120591 +1198911119906V120594+1198911VV120577)

+ 119887lowast

119899(1198912119906119906120591 +1198912119906V120594+1198912VV120577)]

⟨119902lowast 11987611990811119902

⟩ =ℓ120587

4[119886lowast

119899(1198911119906119906120585 +1198911119906V120578 +1198911VV120574)

+ 119887lowast

119899(1198912119906119906120585 +1198912119906V120578 +1198912VV120574)]

+ℓ120587

2[119886lowast

119899(1198911119906119906120591 +1198911119906V120594+1198911VV120577)

+ 119887lowast

119899(1198912119906119906120591 +1198912119906V120594+1198912VV120577)]

(72)

Substituting 119887119899= (1198941205960minus119860(1205820)+1198891119899

2ℓ2)119861(1205820) into (61) gives

Re 119888119896119899

=119891119896119906119906

1198612 minus 2119891119896119906V (119860 minus 1198891119899

2ℓ2) 119861 + 119891119896VV [2 (119860 minus 1198891119899

2ℓ2)2+ 119861119862]

1198612

Im 119888119896119899

=21205960 [119891119896119906V119861 minus 119891

119896VV (119860 minus 11988911198992ℓ2)]

1198612

Journal of Applied Mathematics 9

119889119896119899

=119891119896119906119906

(119860 minus 11988911198992ℓ2) minus 2119891

119896119906V119861 minus 119891119896VV119862

119861

Re 119890119896119899

=119891119896119906119906119906

1198612 minus 3119891119896119906119906V (119860 minus 1198891119899

2ℓ2) 119861 + 2119891119896119906VV (119860 minus 1198891119899

2ℓ2)2+ 119891119896VVV (119860 minus 1198891119899

2ℓ2) 119862

1198612

Im 119890119896119899

=1205960 [119891119896119906119906V119861 minus 2119891

119896119906VV (119860 minus 11988911198992ℓ2) minus 119891

119896VVV119862]

1198612

119896 = 1 2(73)

Then from (70) and (73) we have

Re 120585 =[(119860 + (31198892 minus 1198891) 119899

2ℓ2)Re 1198881119899 + 119861Re 1198882119899 minus 21205960 Im 1198881119899] 1205721 + [(119860 + (31198892 minus 1198891) 119899

2ℓ2) Im 1198881119899 + 119861 Im 1198882119899 + 21205960 Re 1198881119899] 1205722

2 (12057221 + 1205722

2)

Im 120585 =[(119860 + (31198892 minus 1198891) 119899

2ℓ2) Im 1198881119899 + 119861 Im 1198882119899 + 21205960 Re 1198881119899] 1205721 minus [(119860 + (31198892 minus 1198891) 1198992ℓ2)Re 1198881119899 + 119861Re 1198882119899 minus 21205960 Im 1198881119899] 1205722

2 (12057221 + 1205722

2)

Re 120578

=[(119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0]Re 1198881119899 minus 31205960 ((1198891 minus 1198892) 1198992ℓ2 minus 119860) Im 1198881119899 + 119861 (51198891119899

2ℓ2 minus 2119860)Re 1198882119899 minus 1198611205960 Im 1198882119899

2119861 (12057221 + 1205722

2)

sdot 1205721

+[(119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0] Im 1198881119899 + 31205960 ((1198891 minus 1198892) 1198992ℓ2 minus 119860)Re 1198881119899 + 119861 (51198891119899

2ℓ2 minus 2119860) Im 1198882119899 + 1198611205960 Re 11988821198992119861 (1205722

1 + 12057222)

sdot 1205722

Im 120578

=[(119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0] Im 1198881119899 + 31205960 ((1198891 minus 1198892) 1198992ℓ2 minus 119860)Re 1198881119899 + 119861 (51198891119899

2ℓ2 minus 2119860) Im 1198882119899 + 1198611205960 Re 11988821198992119861 (1205722

1 + 12057222)

sdot 1205721

minus[(119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0]Re 1198881119899 minus 31205960 ((1198891 minus 1198892) 1198992ℓ2 minus 119860) Im 1198881119899 + 119861 (51198891119899

2ℓ2 minus 2119860)Re 1198882119899 minus 1198611205960 Im 1198882119899

2119861 (12057221 + 1205722

2)

sdot 1205722

Re 120574 =(1198891119899

2ℓ2 minus 119860) [119862Re 1198881119899 minus (119860 minus 411988911198992ℓ2)Re 1198882119899 minus 21205960 Im 1198882119899] + 1205960 [119862 Im 1198881119899 minus (119860 minus 41198891119899

2ℓ2) Im 1198882119899 + 21205960 Re 1198882119899]2119861 (1205722

1 + 12057222)

1205721

+(1198891119899

2ℓ2 minus 119860) [119862 Im 1198881119899 minus (119860 minus 411988911198992ℓ2) Im 1198882119899 + 21205960 Re 1198882119899] minus 1205960 [119862Re 1198881119899 minus (119860 minus 41198891119899

2ℓ2)Re 1198882119899 minus 21205960 Im 1198882119899]

2119861 (12057221 + 1205722

2)1205722

Im 120574 =(1198891119899

2ℓ2 minus 119860) [119862 Im 1198881119899 minus (119860 minus 411988911198992ℓ2) Im 1198882119899 + 21205960 Re 1198882119899] minus 1205960 [119862Re 1198881119899 minus (119860 minus 41198891119899

2ℓ2)Re 1198882119899 minus 21205960 Im 1198882119899]

2119861 (12057221 + 1205722

2)1205721

10 Journal of Applied Mathematics

minus(1198891119899

2ℓ2 minus 119860) [119862Re 1198881119899 minus (119860 minus 411988911198992ℓ2)Re 1198882119899 minus 21205960 Im 1198882119899] + 1205960 [119862 Im 1198881119899 minus (119860 minus 41198891119899

2ℓ2) Im 1198882119899 + 21205960 Re 1198882119899]2119861 (1205722

1 + 12057222)

1205722

Re 120591 =[(119860 + (1198891 + 1198892) 119899

2ℓ

2)Re 1198881119899 + 119861Re 1198882119899 minus 21205960 Im 1198881119899] 1205723 + [(119860 + (1198891 + 1198892) 119899

2ℓ

2) Im 1198881119899 + 119861 Im 1198882119899 + 21205960 Re 1198881119899] 1205724

2 (12057223 + 1205722

4)

Im 120591 =[(119860 + (1198891 + 1198892) 119899

2ℓ2) Im 1198881119899 + 119861 Im 1198882119899 + 21205960 Re 1198881119899] 1205723 minus [(119860 + (1198891 + 1198892) 1198992ℓ2)Re 1198881119899 + 119861Re 1198882119899 minus 21205960 Im 1198881119899] 1205724

2 (12057223 + 1205722

4)

Re120594

=[(119860 minus (1198891 + 1198892) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0]Re 1198881119899 minus 1205960 ((31198891 + 1198892) 1198992ℓ2 minus 3119860) Im 1198881119899 + 119861 (1198891119899

2ℓ2 minus 2119860)Re 1198882119899 minus 1198611205960 Im 1198882119899

2119861 (12057223 + 1205722

4)

sdot 1205723

+[(119860 minus (1198891 + 1198892) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0] Im 1198881119899 + 1205960 ((31198891 + 1198892) 1198992ℓ2 minus 3119860)Re 1198881119899 + 119861 (1198891119899

2ℓ2 minus 2119860) Im 1198882119899 + 1198611205960 Re 11988821198992119861 (1205722

3 + 12057224)

sdot 1205724

Im120594

=[(119860 minus (1198891 + 1198892) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0] Im 1198881119899 + 1205960 ((31198891 + 1198892) 1198992ℓ2 minus 3119860)Re 1198881119899 + 119861 (1198891119899

2ℓ2 minus 2119860) Im 1198882119899 + 1198611205960 Re 11988821198992119861 (1205722

3 + 12057224)

sdot 1205723

minus[(119860 minus (1198891 + 1198892) 119899

2ℓ

2) (1198891119899

2ℓ

2minus 119860) + 119861119862 + 21205962

0]Re 1198881119899 minus 1205960 ((31198891 + 1198892) 1198992ℓ

2minus 3119860) Im 1198881119899 + 119861 (1198891119899

2ℓ

2minus 2119860)Re 1198882119899 minus 1198611205960 Im 1198882119899

2119861 (12057223 + 1205722

4)

sdot 1205724

Re 120577 =(1198891119899

2ℓ

2minus 119860) (119862Re 1198881119899 minus 119860Re 1198882119899 minus 21205960 Im 1198882119899) + 1205960 (119862 Im 1198881119899 minus 119860 Im 1198882119899 + 21205960 Re 1198882119899)

2119861 (12057223 + 1205722

4)1205723

+(1198891119899

2ℓ2 minus 119860) (119862 Im 1198881119899 minus 119860 Im 1198882119899 + 21205960 Re 1198882119899) minus 1205960 (119862Re 1198881119899 minus 119860Re 1198882119899 minus 21205960 Im 1198882119899)

2119861 (12057223 + 1205722

4)1205724

Im 120577 =(1198891119899

2ℓ2 minus 119860) (119862 Im 1198881119899 minus 119860 Im 1198882119899 + 21205960 Re 1198882119899) minus 1205960 (119862Re 1198881119899 minus 119860Re 1198882119899 minus 21205960 Im 1198882119899)

2119861 (12057223 + 1205722

4)1205723

minus(1198891119899

2ℓ2 minus 119860) (119862Re 1198881119899 minus 119860Re 1198882119899 minus 21205960 Im 1198882119899) + 1205960 (119862 Im 1198881119899 minus 119860 Im 1198882119899 + 21205960 Re 1198882119899)2119861 (1205722

3 + 12057224)

1205724

Re 120578 =[(119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862] 1198891119899 + 119861 (51198891119899

2ℓ2 minus 2119860)1198892119899

21198611205725

Re 120578 =1205960 [(119860 + (31198892 minus 1198891) 119899

2ℓ2) 1198891119899 + 1198611198892119899]

21205725

Re 120574 =(1198891119899

2ℓ2 minus 119860) [1198621198891119899 + (411988911198992ℓ2 minus 119860) 1198892119899]

21198611205725

Im 120574 =1205960 [1198621198891119899 + (41198891119899

2ℓ2 minus 119860) 1198892119899]

21198611205725

Re120594 =[(119860 minus (1198891 + 1198892) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862] 1198891119899 + 119861 (1198891119899

2ℓ2 minus 2119860)1198892119899

21198611205726

Im120594 =1205960 [(119860 minus (1198891 + 1198892) 119899

2ℓ2) 1198891119899 + 1198611198892119899]

21205726119861 (1205820)

Re 120577 =(1198621198891119899 minus 1198601198892119899) (1198891119899

2ℓ

2minus 119860)

21198611205726

Im 120577 =1205960 (1198621198891119899 minus 1198601198892119899)

21198611205726

(74)

Journal of Applied Mathematics 11

Since ℓ120587119886lowast

119899= 1 minus 119894(119860 minus 1198891119899

2ℓ2)1205960 and ℓ120587119887lowast

119899= minus1198941198611205960

one can get

Re ⟨119902lowast 11987611990820119902

⟩ =14[1198911119906119906 (Re 120585 + 2 Re 120591)

+1198911119906V (Re 120578 + 2 Re120594) +1198911VV (Re 120574 + 2 Re 120577)]

+(119860 minus 1198891119899

2ℓ2)

41205960[1198911119906119906 (Im 120585 + 2 Im 120591)

+1198911119906V (Im 120578 + 2 Im120594) +1198911VV (Im 120574 + 2 Im 120577)]

+119861

41205960[1198912119906119906 (Im 120585 + 2 Im 120591)

+1198912119906V (Im 120578 + 2 Im120594) +1198912VV (Im 120574 + 2 Im 120577)]

Re ⟨119902lowast 11987611990811119902

⟩ =14[1198911119906119906 (120585 + 2120591)

+1198911119906V (Re 120578 + 2 Re120594) +1198911VV (Re 120574 + 2 Re 120577)]

+(119860 minus 1198891119899

2ℓ2)

41205960[1198911119906119906 (120585 + 2120591)

+1198911119906V (Re 120578 + 2 Re120594) +1198911VV (Re 120574 + 2 Re 120577)]

+119861

41205960[1198912119906V (Im 120578 + 2 Im120594)

+1198912VV (Im 120574 + 2 Im 120577)]

(75)

In addition it follows from intℓ120587

0 cos4(119899ℓ)119909 119889119909 = 3ℓ1205878and (62) that

⟨119902lowast 119862119902119902119902

⟩ =3ℓ1205878

(119886lowast

1198991198901119899 + 119887

lowast

1198991198902119899) (76)

Therefore

Re ⟨119902lowast 119862119902119902119902

⟩ =381198612 1198911119906119906119906119861

2

minus 31198911119906119906V (119860minus1198891119899

2

ℓ2)119861+ 21198911119906VV (119860minus

11988911198992

ℓ2)

2

+1198911VVV (119860minus1198891119899

2

ℓ2)119862+(119860minus

11988911198992

ℓ2)

sdot [1198911119906119906V119861minus 21198911119906VV (119860minus1198891119899

2

ℓ2)minus1198911VVV119862]

+119861[1198912119906119906V119861minus 21198912119906VV (119860minus1198891119899

2

ℓ2)minus1198912VVV119862]

(77)

Now by (40) we have

Re 1198881 (1205820) = Re ⟨119902lowast 11987611990811119902

⟩+12Re ⟨119902lowast 119876

11990820119902⟩+

12

sdotRe ⟨119902lowast 119862119902119902119902

⟩

=18[1198911119906119906 (Re 120585 + 2 Re 120591 + 2120585 + 4120591)

+1198911119906V (Re 120578 + 2 Re120594+ 2 Re 120578 + 4 Re120594)

+1198911VV (Re 120574 + 2 Re 120577 + 2 Re 120574 + 4 Re 120577)]

+(119860 minus 1198891119899

2ℓ2)

81205960[1198911119906119906 (Im 120585 + 2 Im 120591 + 2120585 + 4120591)

+1198911119906V (Im 120578 + 2 Im120594+ 2 Re 120578 + 4 Re120594)

+1198911VV (Im 120574 + 2 Im 120577 + 2 Re 120574 + 4 Re 120577)]

+119861

81205960[1198912119906119906 (Im 120585 + 2 Im 120591)

+1198912119906V (Im 120578119868+ 2 Im120594+ 2 Im 120578 + 4 Im120594)

+1198912VV (Im 120574 + Im 120577 + 2 Im 120574 + 4 Im 120577)]

+3

161198612 11989111199061199061199061198612minus 31198911119906119906V (119860minus

11988911198992

ℓ2)119861

+ 21198911119906VV (119860minus1198891119899

2

ℓ2)

2

+1198911VVV (119860minus1198891119899

2

ℓ2)119862

+(119860minus1198891119899

2

ℓ2)

sdot [1198911119906119906V119861minus 21198911119906VV (119860minus1198891119899

2

ℓ2)minus1198911VVV119862]

+119861[1198912119906119906V119861minus 21198912119906VV (119860minus1198891119899

2

ℓ2)minus1198912VVV119862]

(78)

Thus we have the following result

Theorem 5 Assume that condition (18) holds for 119899 isin N Thenthe spatially nonhomogeneous Hopf bifurcation of system (1) at(1205820 0 0) is supercritical (resp subcritical) if

Re 1198881 (1205820)

1198601015840 (1205820) + 1198631015840 (1205820)lt 0 (119903119890119904119901 gt 0) (79)

here Re 1198881(1205820) is given by (78)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

12 Journal of Applied Mathematics

Acknowledgments

Cun-Hua Zhang was supported by the Natural ScienceFoundation of Gansu Province (1212RJZA065) Xiang-PingYan was supported by the National Natural Science Foun-dation of China (11261028) Gansu Province Natural ScienceFoundation (145RJZA216) and China Scholarship Council

References

[1] A Andronov E A Leontovich I I Gordon and A G MaierTheory of Bifurcations of Dynamical Systems on a Plane IsraelProgram for Scientific Translations Jerusalem Israel 1971

[2] J Hale Theory of Functional Differential Equations SpringerNew York NY USA 1977

[3] B D Hassard N D Kazarinoff and Y H Wan Theory andApplications of Hopf Bifurcation Cambridge University PressCambridge UK 1981

[4] H Kielhofer Bifurcation Theory An Introduction with Appli-cations to PDEs vol 156 of Applied Mathematical SciencesSpringer New York NY USA 2004

[5] YA KuznetsovElements of Applied BifurcationTheory AppliedMathematical Sciences Springer New York NY USA 1998

[6] J-H Wu Theory and Applications of Partial Functional Differ-ential Equations Springer New York NY USA 1996

[7] F Yi JWei and J Shi ldquoBifurcation and spatiotemporal patternsin a homogeneous diffusive predator-prey systemrdquo Journal ofDifferential Equations vol 246 no 5 pp 1944ndash1977 2009

[8] L-J Yang and X-W Zeng ldquoStability of singular Hopf bifurca-tionsrdquo Journal of Differential Equations vol 206 no 1 pp 30ndash54 2004

[9] J Y Jin J P Shi J J Wei and F Q Yi ldquoBifurcations of patternedsolutions in the diffusive Lengyel-Epstein system of CIMAchemical reactionsrdquo Rocky Mountain Journal of Mathematicsvol 43 no 5 pp 1637ndash1674 2013

[10] S G Ruan ldquoDiffusion-driven instability in the Gierer-Mein-hardtmodel ofmorphogenesisrdquoNatural ResourceModeling vol11 no 2 pp 131ndash142 1998

[11] F Yi J Wei and J Shi ldquoDiffusion-driven instability andbifurcation in the Lengyel-Epstein systemrdquo Nonlinear AnalysisReal World Applications vol 9 no 3 pp 1038ndash1051 2008

[12] F Q Yi J J Wei and J P Shi ldquoGlobal asymptotical behavior ofthe Lengyel-Epstein reaction-diffusion systemrdquoApplied Mathe-matics Letters vol 22 no 1 pp 52ndash55 2009

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 7: Research Article Normal Forms of Hopf Bifurcation for a ...

Journal of Applied Mathematics 7

Let all the second- and third-order partial derivatives of119891119896(120582 119906 V) (119896 = 1 2) with respect to 119906 and V be evaluated at

(1205820 0 0) and let

119888119896119899

= 119891119896119906119906

+ 2119891119896119906V119887119899 +119891

119896VV1198872119899

119889119896119899

= 119891119896119906119906

+119891119896119906V (119887119899 + 119887

119899) +119891119896VV

100381610038161003816100381611988711989910038161003816100381610038162

119890119896119899

= 119891119896119906119906119906

+119891119896119906119906V (2119887119899 + 119887

119899) +119891119896119906VV (2

100381610038161003816100381611988711989910038161003816100381610038162+ 119887

2119899)

+119891119896VVV

100381610038161003816100381611988711989910038161003816100381610038162119887119899

119896 = 1 2

(61)

Then from (30) and (31) one can observe

119876119902119902

= (1198881119899

1198882119899) cos2 119899

ℓ119909 =

12(1198881119899

1198882119899)(1+ cos 2119899

ℓ119909)

119876119902119902

= (1198891119899

1198892119899) cos2 119899

ℓ119909 =

12(1198891119899

1198892119899)(1+ cos 2119899

ℓ119909)

119862119902119902119902

= (1198901119899

1198902119899) cos3 119899

ℓ119909

(62)

In view of (34) (59) and (62) we have

11986720 = 119876119902119902

=12(1198881119899

1198882119899)(1+ cos 2119899

ℓ119909)

11986711 = 119876119902119902

=12(1198891119899

1198892119899)(1+ cos 2119899

ℓ119909)

(63)

Equalities (63) show that the calculation of [21198941205960 minus 119871(1205820)]minus1

and [119871(1205820)]minus1 will be restricted on the subspaces spanned by

eigenmodes 1 and cos(2119899ℓ)119909Let

1205721 =(1211988911198892 minus 31198892

1) 1198994

ℓ4minus3 (1198892 minus 1198891) 1198601198992

ℓ2minus 31205962

0

1205722 =6 (1198891 + 1198892) 119899

21205960ℓ2

1205723 =11988921119899

4

ℓ4+

(1198892 minus 1198891) 1198601198992

ℓ2minus 31205962

0

1205724 = minus2 (1198891 + 1198892) 119899

21205960ℓ2

(64)

Then under condition (18) one can derive

[21198941205960119868 minus 1198712119899 (1205820)]minus1

=1

1205721 + 1198941205722

sdot(

21198941205960 + 119860 +(31198892 minus 1198891) 119899

2

ℓ2119861

119862 21198941205960 minus 119860 +41198891119899

2

ℓ2

)

[21198941205960119868 minus 1198710 (1205820)]minus1

=1

1205723 + 1198941205723

sdot(21198941205960 + 119860 minus

(1198891 + 1198892) 1198992

ℓ2119861

119862 21198941205960 minus 119860

)

(65)

From (36) and (61) we have

11990820 = [21198941205960119868 minus 1198712119899 (1205820)]

minus1

2cos 2119899

ℓ119909

+[21198941205960119868 minus 1198710 (1205820)]

minus1

2(

1198881119899

1198882119899) =

12 (1205721 + 1198941205722)

sdot(

[21198941205960 + 119860 +(31198892 minus 1198891) 119899

2

ℓ2] 1198881119899 + 1198611198882119899

1198621198881119899 + (21198941205960 minus 119860 +41198891119899

2

ℓ2) 1198882119899

)

sdot cos 2119899ℓ

119909 +1

2 (1205723 + 1198941205724)

sdot([21198941205960 + 119860 minus

(1198891 + 1198892) 1198992

ℓ2] 1198881119899 + 1198611198882119899

1198621198881119899 + (21198941205960 minus 119860) 1198882119899

)

(66)

Similarly we can get

11990811 =121205725

(

[119860 +(31198892 minus 1198891) 119899

2

ℓ2]1198891119899 + 1198611198892119899

1198621198891119899 + (41198891119899

2

ℓ2minus 119860)1198892119899

)

sdot cos 2119899ℓ

119909 +121205726

sdot([119860 minus

(1198891 + 1198892) 1198992

ℓ2]1198891119899 + 1198611198892119899

1198621198891119899 minus 1198601198892119899

)

(67)

where

1205725 =(1211988911198892 minus 31198892

1) 1198994

ℓ4minus3 (1198892 minus 1198891) 119860119899

2

ℓ2+120596

20

1205726 =11988921119899

4

ℓ4+

(1198892 minus 1198891) 1198601198992

ℓ2+120596

20

(68)

8 Journal of Applied Mathematics

From (30) we have

11987611990820119902

= (1198911119906119906120585 + 1198911119906V120578 + 1198911VV120574

1198912119906119906120585 + 1198912119906V120578 + 1198912VV120574) cos 119899

ℓ119909 cos 2119899

ℓ119909

+(1198911119906119906120591 + 1198911119906V120594 + 1198911VV120577

1198912119906119906120591 + 1198912119906V120594 + 1198912VV120577) cos 119899

ℓ119909

11987611990811119902

= (1198911119906119906120585 + 1198911119906V120578 + 1198911VV120574

1198912119906119906120585 + 1198912119906V120578 + 1198912VV120574) cos 119899

ℓ119909 cos 2119899

ℓ119909

+(1198911119906119906120591 + 1198911119906V120594 + 1198911VV120577

1198912119906119906120591 + 1198912119906V120594 + 1198912VV120577) cos 119899

ℓ119909

(69)

with

120585 =(21198941205960 + 119860 + (31198892 minus 1198891) 119899

2ℓ

2) 1198881119899 + 1198611198882119899

2 (1205721 + 1198941205722)

120578 =(21198941205960 + 119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860 minus 1198941205960) + 119861119862

2119861 (1205721 + 1198941205722)

sdot 1198881119899 +(51198891119899

2ℓ2 minus 2119860 + 1198941205960)

2 (1205721 + 1198941205722)1198882119899

120574 =[1198621198881119899 + (21198941205960 minus 119860 + 41198891119899

2ℓ2) 1198882119899] (11988911198992ℓ2 minus 119860 minus 1198941205960)

2119861 (1205721 + 1198941205722)

120591 =(21198941205960 + 119860 minus (1198891 + 1198892) 119899

2ℓ2) 1198881119899 + 1198611198882119899

2 (1205723 + 1198941205724)

120594 =(21198941205960 + 119860 minus (1198891 + 1198892) 119899

2ℓ2) (11988911198992ℓ2 minus 119860 minus 1198941205960) + 119861119862

2119861 (1205723 + 1198941205724)

sdot 1198881119899 +(1198891119899

2ℓ2 minus 2119860 + 1198941205960)

2 (1205723 + 1198941205724)1198882119899

120577 =[1198621198881119899 + (21198941205960 minus 119860) 1198882119899] (1198891119899

2ℓ2 minus 119860 minus 1198941205960)

2119861 (1205723 + 1198941205724)

120585 =(119860 + (31198892 minus 1198891) 119899

2ℓ2) 1198891119899 + 1198611198892119899

21205725

120578 =(119860 + (31198892 minus 1198891) 119899

2ℓ2) (1198941205960 minus 119860 + 11988911198992ℓ2) + 119861119862

211986112057251198891119899

+(1198941205960 minus 2119860 + 51198891119899

2ℓ2)

212057251198892119899

120574 =(1198941205960 minus 119860 + 1198891119899

2ℓ2) [1198621198891119899 + (411988911198992ℓ2 minus 119860) 1198892119899]

21198611205725

120591 =(119860 minus (1198891 + 1198892) 119899

2ℓ2) 1198891119899 + 1198611198892119899

21205726

120594 =(119860 minus (1198891 + 1198892) 119899

2ℓ2) (1198941205960 minus 119860 + 11988911198992ℓ2) + 119861119862

211986112057261198891119899

+(1198941205960 minus 2119860 + 1198891119899

2ℓ2)

212057261198892119899

120577 =(1198941205960 minus 119860 + 1198891119899

2ℓ2) (1198621198891119899 minus 1198601198892119899)

21198611205726

(70)

Notice that for 119899 isin N

intℓ120587

0cos2 119899

ℓ119909 119889119909 =

ℓ120587

2

intℓ120587

0cos 2119899

ℓ119909cos2 119899

ℓ119909 119889119909 =

ℓ120587

4

(71)

It follows from (69) that

⟨119902lowast 11987611990820119902

⟩ =ℓ120587

4[119886lowast

119899(1198911119906119906120585 +1198911119906V120578 +1198911VV120574)

+ 119887lowast

119899(1198912119906119906120585 +1198912119906V120578 +1198912VV120574)]

+ℓ120587

2[119886lowast

119899(1198911119906119906120591 +1198911119906V120594+1198911VV120577)

+ 119887lowast

119899(1198912119906119906120591 +1198912119906V120594+1198912VV120577)]

⟨119902lowast 11987611990811119902

⟩ =ℓ120587

4[119886lowast

119899(1198911119906119906120585 +1198911119906V120578 +1198911VV120574)

+ 119887lowast

119899(1198912119906119906120585 +1198912119906V120578 +1198912VV120574)]

+ℓ120587

2[119886lowast

119899(1198911119906119906120591 +1198911119906V120594+1198911VV120577)

+ 119887lowast

119899(1198912119906119906120591 +1198912119906V120594+1198912VV120577)]

(72)

Substituting 119887119899= (1198941205960minus119860(1205820)+1198891119899

2ℓ2)119861(1205820) into (61) gives

Re 119888119896119899

=119891119896119906119906

1198612 minus 2119891119896119906V (119860 minus 1198891119899

2ℓ2) 119861 + 119891119896VV [2 (119860 minus 1198891119899

2ℓ2)2+ 119861119862]

1198612

Im 119888119896119899

=21205960 [119891119896119906V119861 minus 119891

119896VV (119860 minus 11988911198992ℓ2)]

1198612

Journal of Applied Mathematics 9

119889119896119899

=119891119896119906119906

(119860 minus 11988911198992ℓ2) minus 2119891

119896119906V119861 minus 119891119896VV119862

119861

Re 119890119896119899

=119891119896119906119906119906

1198612 minus 3119891119896119906119906V (119860 minus 1198891119899

2ℓ2) 119861 + 2119891119896119906VV (119860 minus 1198891119899

2ℓ2)2+ 119891119896VVV (119860 minus 1198891119899

2ℓ2) 119862

1198612

Im 119890119896119899

=1205960 [119891119896119906119906V119861 minus 2119891

119896119906VV (119860 minus 11988911198992ℓ2) minus 119891

119896VVV119862]

1198612

119896 = 1 2(73)

Then from (70) and (73) we have

Re 120585 =[(119860 + (31198892 minus 1198891) 119899

2ℓ2)Re 1198881119899 + 119861Re 1198882119899 minus 21205960 Im 1198881119899] 1205721 + [(119860 + (31198892 minus 1198891) 119899

2ℓ2) Im 1198881119899 + 119861 Im 1198882119899 + 21205960 Re 1198881119899] 1205722

2 (12057221 + 1205722

2)

Im 120585 =[(119860 + (31198892 minus 1198891) 119899

2ℓ2) Im 1198881119899 + 119861 Im 1198882119899 + 21205960 Re 1198881119899] 1205721 minus [(119860 + (31198892 minus 1198891) 1198992ℓ2)Re 1198881119899 + 119861Re 1198882119899 minus 21205960 Im 1198881119899] 1205722

2 (12057221 + 1205722

2)

Re 120578

=[(119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0]Re 1198881119899 minus 31205960 ((1198891 minus 1198892) 1198992ℓ2 minus 119860) Im 1198881119899 + 119861 (51198891119899

2ℓ2 minus 2119860)Re 1198882119899 minus 1198611205960 Im 1198882119899

2119861 (12057221 + 1205722

2)

sdot 1205721

+[(119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0] Im 1198881119899 + 31205960 ((1198891 minus 1198892) 1198992ℓ2 minus 119860)Re 1198881119899 + 119861 (51198891119899

2ℓ2 minus 2119860) Im 1198882119899 + 1198611205960 Re 11988821198992119861 (1205722

1 + 12057222)

sdot 1205722

Im 120578

=[(119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0] Im 1198881119899 + 31205960 ((1198891 minus 1198892) 1198992ℓ2 minus 119860)Re 1198881119899 + 119861 (51198891119899

2ℓ2 minus 2119860) Im 1198882119899 + 1198611205960 Re 11988821198992119861 (1205722

1 + 12057222)

sdot 1205721

minus[(119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0]Re 1198881119899 minus 31205960 ((1198891 minus 1198892) 1198992ℓ2 minus 119860) Im 1198881119899 + 119861 (51198891119899

2ℓ2 minus 2119860)Re 1198882119899 minus 1198611205960 Im 1198882119899

2119861 (12057221 + 1205722

2)

sdot 1205722

Re 120574 =(1198891119899

2ℓ2 minus 119860) [119862Re 1198881119899 minus (119860 minus 411988911198992ℓ2)Re 1198882119899 minus 21205960 Im 1198882119899] + 1205960 [119862 Im 1198881119899 minus (119860 minus 41198891119899

2ℓ2) Im 1198882119899 + 21205960 Re 1198882119899]2119861 (1205722

1 + 12057222)

1205721

+(1198891119899

2ℓ2 minus 119860) [119862 Im 1198881119899 minus (119860 minus 411988911198992ℓ2) Im 1198882119899 + 21205960 Re 1198882119899] minus 1205960 [119862Re 1198881119899 minus (119860 minus 41198891119899

2ℓ2)Re 1198882119899 minus 21205960 Im 1198882119899]

2119861 (12057221 + 1205722

2)1205722

Im 120574 =(1198891119899

2ℓ2 minus 119860) [119862 Im 1198881119899 minus (119860 minus 411988911198992ℓ2) Im 1198882119899 + 21205960 Re 1198882119899] minus 1205960 [119862Re 1198881119899 minus (119860 minus 41198891119899

2ℓ2)Re 1198882119899 minus 21205960 Im 1198882119899]

2119861 (12057221 + 1205722

2)1205721

10 Journal of Applied Mathematics

minus(1198891119899

2ℓ2 minus 119860) [119862Re 1198881119899 minus (119860 minus 411988911198992ℓ2)Re 1198882119899 minus 21205960 Im 1198882119899] + 1205960 [119862 Im 1198881119899 minus (119860 minus 41198891119899

2ℓ2) Im 1198882119899 + 21205960 Re 1198882119899]2119861 (1205722

1 + 12057222)

1205722

Re 120591 =[(119860 + (1198891 + 1198892) 119899

2ℓ

2)Re 1198881119899 + 119861Re 1198882119899 minus 21205960 Im 1198881119899] 1205723 + [(119860 + (1198891 + 1198892) 119899

2ℓ

2) Im 1198881119899 + 119861 Im 1198882119899 + 21205960 Re 1198881119899] 1205724

2 (12057223 + 1205722

4)

Im 120591 =[(119860 + (1198891 + 1198892) 119899

2ℓ2) Im 1198881119899 + 119861 Im 1198882119899 + 21205960 Re 1198881119899] 1205723 minus [(119860 + (1198891 + 1198892) 1198992ℓ2)Re 1198881119899 + 119861Re 1198882119899 minus 21205960 Im 1198881119899] 1205724

2 (12057223 + 1205722

4)

Re120594

=[(119860 minus (1198891 + 1198892) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0]Re 1198881119899 minus 1205960 ((31198891 + 1198892) 1198992ℓ2 minus 3119860) Im 1198881119899 + 119861 (1198891119899

2ℓ2 minus 2119860)Re 1198882119899 minus 1198611205960 Im 1198882119899

2119861 (12057223 + 1205722

4)

sdot 1205723

+[(119860 minus (1198891 + 1198892) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0] Im 1198881119899 + 1205960 ((31198891 + 1198892) 1198992ℓ2 minus 3119860)Re 1198881119899 + 119861 (1198891119899

2ℓ2 minus 2119860) Im 1198882119899 + 1198611205960 Re 11988821198992119861 (1205722

3 + 12057224)

sdot 1205724

Im120594

=[(119860 minus (1198891 + 1198892) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0] Im 1198881119899 + 1205960 ((31198891 + 1198892) 1198992ℓ2 minus 3119860)Re 1198881119899 + 119861 (1198891119899

2ℓ2 minus 2119860) Im 1198882119899 + 1198611205960 Re 11988821198992119861 (1205722

3 + 12057224)

sdot 1205723

minus[(119860 minus (1198891 + 1198892) 119899

2ℓ

2) (1198891119899

2ℓ

2minus 119860) + 119861119862 + 21205962

0]Re 1198881119899 minus 1205960 ((31198891 + 1198892) 1198992ℓ

2minus 3119860) Im 1198881119899 + 119861 (1198891119899

2ℓ

2minus 2119860)Re 1198882119899 minus 1198611205960 Im 1198882119899

2119861 (12057223 + 1205722

4)

sdot 1205724

Re 120577 =(1198891119899

2ℓ

2minus 119860) (119862Re 1198881119899 minus 119860Re 1198882119899 minus 21205960 Im 1198882119899) + 1205960 (119862 Im 1198881119899 minus 119860 Im 1198882119899 + 21205960 Re 1198882119899)

2119861 (12057223 + 1205722

4)1205723

+(1198891119899

2ℓ2 minus 119860) (119862 Im 1198881119899 minus 119860 Im 1198882119899 + 21205960 Re 1198882119899) minus 1205960 (119862Re 1198881119899 minus 119860Re 1198882119899 minus 21205960 Im 1198882119899)

2119861 (12057223 + 1205722

4)1205724

Im 120577 =(1198891119899

2ℓ2 minus 119860) (119862 Im 1198881119899 minus 119860 Im 1198882119899 + 21205960 Re 1198882119899) minus 1205960 (119862Re 1198881119899 minus 119860Re 1198882119899 minus 21205960 Im 1198882119899)

2119861 (12057223 + 1205722

4)1205723

minus(1198891119899

2ℓ2 minus 119860) (119862Re 1198881119899 minus 119860Re 1198882119899 minus 21205960 Im 1198882119899) + 1205960 (119862 Im 1198881119899 minus 119860 Im 1198882119899 + 21205960 Re 1198882119899)2119861 (1205722

3 + 12057224)

1205724

Re 120578 =[(119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862] 1198891119899 + 119861 (51198891119899

2ℓ2 minus 2119860)1198892119899

21198611205725

Re 120578 =1205960 [(119860 + (31198892 minus 1198891) 119899

2ℓ2) 1198891119899 + 1198611198892119899]

21205725

Re 120574 =(1198891119899

2ℓ2 minus 119860) [1198621198891119899 + (411988911198992ℓ2 minus 119860) 1198892119899]

21198611205725

Im 120574 =1205960 [1198621198891119899 + (41198891119899

2ℓ2 minus 119860) 1198892119899]

21198611205725

Re120594 =[(119860 minus (1198891 + 1198892) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862] 1198891119899 + 119861 (1198891119899

2ℓ2 minus 2119860)1198892119899

21198611205726

Im120594 =1205960 [(119860 minus (1198891 + 1198892) 119899

2ℓ2) 1198891119899 + 1198611198892119899]

21205726119861 (1205820)

Re 120577 =(1198621198891119899 minus 1198601198892119899) (1198891119899

2ℓ

2minus 119860)

21198611205726

Im 120577 =1205960 (1198621198891119899 minus 1198601198892119899)

21198611205726

(74)

Journal of Applied Mathematics 11

Since ℓ120587119886lowast

119899= 1 minus 119894(119860 minus 1198891119899

2ℓ2)1205960 and ℓ120587119887lowast

119899= minus1198941198611205960

one can get

Re ⟨119902lowast 11987611990820119902

⟩ =14[1198911119906119906 (Re 120585 + 2 Re 120591)

+1198911119906V (Re 120578 + 2 Re120594) +1198911VV (Re 120574 + 2 Re 120577)]

+(119860 minus 1198891119899

2ℓ2)

41205960[1198911119906119906 (Im 120585 + 2 Im 120591)

+1198911119906V (Im 120578 + 2 Im120594) +1198911VV (Im 120574 + 2 Im 120577)]

+119861

41205960[1198912119906119906 (Im 120585 + 2 Im 120591)

+1198912119906V (Im 120578 + 2 Im120594) +1198912VV (Im 120574 + 2 Im 120577)]

Re ⟨119902lowast 11987611990811119902

⟩ =14[1198911119906119906 (120585 + 2120591)

+1198911119906V (Re 120578 + 2 Re120594) +1198911VV (Re 120574 + 2 Re 120577)]

+(119860 minus 1198891119899

2ℓ2)

41205960[1198911119906119906 (120585 + 2120591)

+1198911119906V (Re 120578 + 2 Re120594) +1198911VV (Re 120574 + 2 Re 120577)]

+119861

41205960[1198912119906V (Im 120578 + 2 Im120594)

+1198912VV (Im 120574 + 2 Im 120577)]

(75)

In addition it follows from intℓ120587

0 cos4(119899ℓ)119909 119889119909 = 3ℓ1205878and (62) that

⟨119902lowast 119862119902119902119902

⟩ =3ℓ1205878

(119886lowast

1198991198901119899 + 119887

lowast

1198991198902119899) (76)

Therefore

Re ⟨119902lowast 119862119902119902119902

⟩ =381198612 1198911119906119906119906119861

2

minus 31198911119906119906V (119860minus1198891119899

2

ℓ2)119861+ 21198911119906VV (119860minus

11988911198992

ℓ2)

2

+1198911VVV (119860minus1198891119899

2

ℓ2)119862+(119860minus

11988911198992

ℓ2)

sdot [1198911119906119906V119861minus 21198911119906VV (119860minus1198891119899

2

ℓ2)minus1198911VVV119862]

+119861[1198912119906119906V119861minus 21198912119906VV (119860minus1198891119899

2

ℓ2)minus1198912VVV119862]

(77)

Now by (40) we have

Re 1198881 (1205820) = Re ⟨119902lowast 11987611990811119902

⟩+12Re ⟨119902lowast 119876

11990820119902⟩+

12

sdotRe ⟨119902lowast 119862119902119902119902

⟩

=18[1198911119906119906 (Re 120585 + 2 Re 120591 + 2120585 + 4120591)

+1198911119906V (Re 120578 + 2 Re120594+ 2 Re 120578 + 4 Re120594)

+1198911VV (Re 120574 + 2 Re 120577 + 2 Re 120574 + 4 Re 120577)]

+(119860 minus 1198891119899

2ℓ2)

81205960[1198911119906119906 (Im 120585 + 2 Im 120591 + 2120585 + 4120591)

+1198911119906V (Im 120578 + 2 Im120594+ 2 Re 120578 + 4 Re120594)

+1198911VV (Im 120574 + 2 Im 120577 + 2 Re 120574 + 4 Re 120577)]

+119861

81205960[1198912119906119906 (Im 120585 + 2 Im 120591)

+1198912119906V (Im 120578119868+ 2 Im120594+ 2 Im 120578 + 4 Im120594)

+1198912VV (Im 120574 + Im 120577 + 2 Im 120574 + 4 Im 120577)]

+3

161198612 11989111199061199061199061198612minus 31198911119906119906V (119860minus

11988911198992

ℓ2)119861

+ 21198911119906VV (119860minus1198891119899

2

ℓ2)

2

+1198911VVV (119860minus1198891119899

2

ℓ2)119862

+(119860minus1198891119899

2

ℓ2)

sdot [1198911119906119906V119861minus 21198911119906VV (119860minus1198891119899

2

ℓ2)minus1198911VVV119862]

+119861[1198912119906119906V119861minus 21198912119906VV (119860minus1198891119899

2

ℓ2)minus1198912VVV119862]

(78)

Thus we have the following result

Theorem 5 Assume that condition (18) holds for 119899 isin N Thenthe spatially nonhomogeneous Hopf bifurcation of system (1) at(1205820 0 0) is supercritical (resp subcritical) if

Re 1198881 (1205820)

1198601015840 (1205820) + 1198631015840 (1205820)lt 0 (119903119890119904119901 gt 0) (79)

here Re 1198881(1205820) is given by (78)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

12 Journal of Applied Mathematics

Acknowledgments

Cun-Hua Zhang was supported by the Natural ScienceFoundation of Gansu Province (1212RJZA065) Xiang-PingYan was supported by the National Natural Science Foun-dation of China (11261028) Gansu Province Natural ScienceFoundation (145RJZA216) and China Scholarship Council

References

[1] A Andronov E A Leontovich I I Gordon and A G MaierTheory of Bifurcations of Dynamical Systems on a Plane IsraelProgram for Scientific Translations Jerusalem Israel 1971

[2] J Hale Theory of Functional Differential Equations SpringerNew York NY USA 1977

[3] B D Hassard N D Kazarinoff and Y H Wan Theory andApplications of Hopf Bifurcation Cambridge University PressCambridge UK 1981

[4] H Kielhofer Bifurcation Theory An Introduction with Appli-cations to PDEs vol 156 of Applied Mathematical SciencesSpringer New York NY USA 2004

[5] YA KuznetsovElements of Applied BifurcationTheory AppliedMathematical Sciences Springer New York NY USA 1998

[6] J-H Wu Theory and Applications of Partial Functional Differ-ential Equations Springer New York NY USA 1996

[7] F Yi JWei and J Shi ldquoBifurcation and spatiotemporal patternsin a homogeneous diffusive predator-prey systemrdquo Journal ofDifferential Equations vol 246 no 5 pp 1944ndash1977 2009

[8] L-J Yang and X-W Zeng ldquoStability of singular Hopf bifurca-tionsrdquo Journal of Differential Equations vol 206 no 1 pp 30ndash54 2004

[9] J Y Jin J P Shi J J Wei and F Q Yi ldquoBifurcations of patternedsolutions in the diffusive Lengyel-Epstein system of CIMAchemical reactionsrdquo Rocky Mountain Journal of Mathematicsvol 43 no 5 pp 1637ndash1674 2013

[10] S G Ruan ldquoDiffusion-driven instability in the Gierer-Mein-hardtmodel ofmorphogenesisrdquoNatural ResourceModeling vol11 no 2 pp 131ndash142 1998

[11] F Yi J Wei and J Shi ldquoDiffusion-driven instability andbifurcation in the Lengyel-Epstein systemrdquo Nonlinear AnalysisReal World Applications vol 9 no 3 pp 1038ndash1051 2008

[12] F Q Yi J J Wei and J P Shi ldquoGlobal asymptotical behavior ofthe Lengyel-Epstein reaction-diffusion systemrdquoApplied Mathe-matics Letters vol 22 no 1 pp 52ndash55 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 8: Research Article Normal Forms of Hopf Bifurcation for a ...

8 Journal of Applied Mathematics

From (30) we have

11987611990820119902

= (1198911119906119906120585 + 1198911119906V120578 + 1198911VV120574

1198912119906119906120585 + 1198912119906V120578 + 1198912VV120574) cos 119899

ℓ119909 cos 2119899

ℓ119909

+(1198911119906119906120591 + 1198911119906V120594 + 1198911VV120577

1198912119906119906120591 + 1198912119906V120594 + 1198912VV120577) cos 119899

ℓ119909

11987611990811119902

= (1198911119906119906120585 + 1198911119906V120578 + 1198911VV120574

1198912119906119906120585 + 1198912119906V120578 + 1198912VV120574) cos 119899

ℓ119909 cos 2119899

ℓ119909

+(1198911119906119906120591 + 1198911119906V120594 + 1198911VV120577

1198912119906119906120591 + 1198912119906V120594 + 1198912VV120577) cos 119899

ℓ119909

(69)

with

120585 =(21198941205960 + 119860 + (31198892 minus 1198891) 119899

2ℓ

2) 1198881119899 + 1198611198882119899

2 (1205721 + 1198941205722)

120578 =(21198941205960 + 119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860 minus 1198941205960) + 119861119862

2119861 (1205721 + 1198941205722)

sdot 1198881119899 +(51198891119899

2ℓ2 minus 2119860 + 1198941205960)

2 (1205721 + 1198941205722)1198882119899

120574 =[1198621198881119899 + (21198941205960 minus 119860 + 41198891119899

2ℓ2) 1198882119899] (11988911198992ℓ2 minus 119860 minus 1198941205960)

2119861 (1205721 + 1198941205722)

120591 =(21198941205960 + 119860 minus (1198891 + 1198892) 119899

2ℓ2) 1198881119899 + 1198611198882119899

2 (1205723 + 1198941205724)

120594 =(21198941205960 + 119860 minus (1198891 + 1198892) 119899

2ℓ2) (11988911198992ℓ2 minus 119860 minus 1198941205960) + 119861119862

2119861 (1205723 + 1198941205724)

sdot 1198881119899 +(1198891119899

2ℓ2 minus 2119860 + 1198941205960)

2 (1205723 + 1198941205724)1198882119899

120577 =[1198621198881119899 + (21198941205960 minus 119860) 1198882119899] (1198891119899

2ℓ2 minus 119860 minus 1198941205960)

2119861 (1205723 + 1198941205724)

120585 =(119860 + (31198892 minus 1198891) 119899

2ℓ2) 1198891119899 + 1198611198892119899

21205725

120578 =(119860 + (31198892 minus 1198891) 119899

2ℓ2) (1198941205960 minus 119860 + 11988911198992ℓ2) + 119861119862

211986112057251198891119899

+(1198941205960 minus 2119860 + 51198891119899

2ℓ2)

212057251198892119899

120574 =(1198941205960 minus 119860 + 1198891119899

2ℓ2) [1198621198891119899 + (411988911198992ℓ2 minus 119860) 1198892119899]

21198611205725

120591 =(119860 minus (1198891 + 1198892) 119899

2ℓ2) 1198891119899 + 1198611198892119899

21205726

120594 =(119860 minus (1198891 + 1198892) 119899

2ℓ2) (1198941205960 minus 119860 + 11988911198992ℓ2) + 119861119862

211986112057261198891119899

+(1198941205960 minus 2119860 + 1198891119899

2ℓ2)

212057261198892119899

120577 =(1198941205960 minus 119860 + 1198891119899

2ℓ2) (1198621198891119899 minus 1198601198892119899)

21198611205726

(70)

Notice that for 119899 isin N

intℓ120587

0cos2 119899

ℓ119909 119889119909 =

ℓ120587

2

intℓ120587

0cos 2119899

ℓ119909cos2 119899

ℓ119909 119889119909 =

ℓ120587

4

(71)

It follows from (69) that

⟨119902lowast 11987611990820119902

⟩ =ℓ120587

4[119886lowast

119899(1198911119906119906120585 +1198911119906V120578 +1198911VV120574)

+ 119887lowast

119899(1198912119906119906120585 +1198912119906V120578 +1198912VV120574)]

+ℓ120587

2[119886lowast

119899(1198911119906119906120591 +1198911119906V120594+1198911VV120577)

+ 119887lowast

119899(1198912119906119906120591 +1198912119906V120594+1198912VV120577)]

⟨119902lowast 11987611990811119902

⟩ =ℓ120587

4[119886lowast

119899(1198911119906119906120585 +1198911119906V120578 +1198911VV120574)

+ 119887lowast

119899(1198912119906119906120585 +1198912119906V120578 +1198912VV120574)]

+ℓ120587

2[119886lowast

119899(1198911119906119906120591 +1198911119906V120594+1198911VV120577)

+ 119887lowast

119899(1198912119906119906120591 +1198912119906V120594+1198912VV120577)]

(72)

Substituting 119887119899= (1198941205960minus119860(1205820)+1198891119899

2ℓ2)119861(1205820) into (61) gives

Re 119888119896119899

=119891119896119906119906

1198612 minus 2119891119896119906V (119860 minus 1198891119899

2ℓ2) 119861 + 119891119896VV [2 (119860 minus 1198891119899

2ℓ2)2+ 119861119862]

1198612

Im 119888119896119899

=21205960 [119891119896119906V119861 minus 119891

119896VV (119860 minus 11988911198992ℓ2)]

1198612

Journal of Applied Mathematics 9

119889119896119899

=119891119896119906119906

(119860 minus 11988911198992ℓ2) minus 2119891

119896119906V119861 minus 119891119896VV119862

119861

Re 119890119896119899

=119891119896119906119906119906

1198612 minus 3119891119896119906119906V (119860 minus 1198891119899

2ℓ2) 119861 + 2119891119896119906VV (119860 minus 1198891119899

2ℓ2)2+ 119891119896VVV (119860 minus 1198891119899

2ℓ2) 119862

1198612

Im 119890119896119899

=1205960 [119891119896119906119906V119861 minus 2119891

119896119906VV (119860 minus 11988911198992ℓ2) minus 119891

119896VVV119862]

1198612

119896 = 1 2(73)

Then from (70) and (73) we have

Re 120585 =[(119860 + (31198892 minus 1198891) 119899

2ℓ2)Re 1198881119899 + 119861Re 1198882119899 minus 21205960 Im 1198881119899] 1205721 + [(119860 + (31198892 minus 1198891) 119899

2ℓ2) Im 1198881119899 + 119861 Im 1198882119899 + 21205960 Re 1198881119899] 1205722

2 (12057221 + 1205722

2)

Im 120585 =[(119860 + (31198892 minus 1198891) 119899

2ℓ2) Im 1198881119899 + 119861 Im 1198882119899 + 21205960 Re 1198881119899] 1205721 minus [(119860 + (31198892 minus 1198891) 1198992ℓ2)Re 1198881119899 + 119861Re 1198882119899 minus 21205960 Im 1198881119899] 1205722

2 (12057221 + 1205722

2)

Re 120578

=[(119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0]Re 1198881119899 minus 31205960 ((1198891 minus 1198892) 1198992ℓ2 minus 119860) Im 1198881119899 + 119861 (51198891119899

2ℓ2 minus 2119860)Re 1198882119899 minus 1198611205960 Im 1198882119899

2119861 (12057221 + 1205722

2)

sdot 1205721

+[(119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0] Im 1198881119899 + 31205960 ((1198891 minus 1198892) 1198992ℓ2 minus 119860)Re 1198881119899 + 119861 (51198891119899

2ℓ2 minus 2119860) Im 1198882119899 + 1198611205960 Re 11988821198992119861 (1205722

1 + 12057222)

sdot 1205722

Im 120578

=[(119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0] Im 1198881119899 + 31205960 ((1198891 minus 1198892) 1198992ℓ2 minus 119860)Re 1198881119899 + 119861 (51198891119899

2ℓ2 minus 2119860) Im 1198882119899 + 1198611205960 Re 11988821198992119861 (1205722

1 + 12057222)

sdot 1205721

minus[(119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0]Re 1198881119899 minus 31205960 ((1198891 minus 1198892) 1198992ℓ2 minus 119860) Im 1198881119899 + 119861 (51198891119899

2ℓ2 minus 2119860)Re 1198882119899 minus 1198611205960 Im 1198882119899

2119861 (12057221 + 1205722

2)

sdot 1205722

Re 120574 =(1198891119899

2ℓ2 minus 119860) [119862Re 1198881119899 minus (119860 minus 411988911198992ℓ2)Re 1198882119899 minus 21205960 Im 1198882119899] + 1205960 [119862 Im 1198881119899 minus (119860 minus 41198891119899

2ℓ2) Im 1198882119899 + 21205960 Re 1198882119899]2119861 (1205722

1 + 12057222)

1205721

+(1198891119899

2ℓ2 minus 119860) [119862 Im 1198881119899 minus (119860 minus 411988911198992ℓ2) Im 1198882119899 + 21205960 Re 1198882119899] minus 1205960 [119862Re 1198881119899 minus (119860 minus 41198891119899

2ℓ2)Re 1198882119899 minus 21205960 Im 1198882119899]

2119861 (12057221 + 1205722

2)1205722

Im 120574 =(1198891119899

2ℓ2 minus 119860) [119862 Im 1198881119899 minus (119860 minus 411988911198992ℓ2) Im 1198882119899 + 21205960 Re 1198882119899] minus 1205960 [119862Re 1198881119899 minus (119860 minus 41198891119899

2ℓ2)Re 1198882119899 minus 21205960 Im 1198882119899]

2119861 (12057221 + 1205722

2)1205721

10 Journal of Applied Mathematics

minus(1198891119899

2ℓ2 minus 119860) [119862Re 1198881119899 minus (119860 minus 411988911198992ℓ2)Re 1198882119899 minus 21205960 Im 1198882119899] + 1205960 [119862 Im 1198881119899 minus (119860 minus 41198891119899

2ℓ2) Im 1198882119899 + 21205960 Re 1198882119899]2119861 (1205722

1 + 12057222)

1205722

Re 120591 =[(119860 + (1198891 + 1198892) 119899

2ℓ

2)Re 1198881119899 + 119861Re 1198882119899 minus 21205960 Im 1198881119899] 1205723 + [(119860 + (1198891 + 1198892) 119899

2ℓ

2) Im 1198881119899 + 119861 Im 1198882119899 + 21205960 Re 1198881119899] 1205724

2 (12057223 + 1205722

4)

Im 120591 =[(119860 + (1198891 + 1198892) 119899

2ℓ2) Im 1198881119899 + 119861 Im 1198882119899 + 21205960 Re 1198881119899] 1205723 minus [(119860 + (1198891 + 1198892) 1198992ℓ2)Re 1198881119899 + 119861Re 1198882119899 minus 21205960 Im 1198881119899] 1205724

2 (12057223 + 1205722

4)

Re120594

=[(119860 minus (1198891 + 1198892) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0]Re 1198881119899 minus 1205960 ((31198891 + 1198892) 1198992ℓ2 minus 3119860) Im 1198881119899 + 119861 (1198891119899

2ℓ2 minus 2119860)Re 1198882119899 minus 1198611205960 Im 1198882119899

2119861 (12057223 + 1205722

4)

sdot 1205723

+[(119860 minus (1198891 + 1198892) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0] Im 1198881119899 + 1205960 ((31198891 + 1198892) 1198992ℓ2 minus 3119860)Re 1198881119899 + 119861 (1198891119899

2ℓ2 minus 2119860) Im 1198882119899 + 1198611205960 Re 11988821198992119861 (1205722

3 + 12057224)

sdot 1205724

Im120594

=[(119860 minus (1198891 + 1198892) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0] Im 1198881119899 + 1205960 ((31198891 + 1198892) 1198992ℓ2 minus 3119860)Re 1198881119899 + 119861 (1198891119899

2ℓ2 minus 2119860) Im 1198882119899 + 1198611205960 Re 11988821198992119861 (1205722

3 + 12057224)

sdot 1205723

minus[(119860 minus (1198891 + 1198892) 119899

2ℓ

2) (1198891119899

2ℓ

2minus 119860) + 119861119862 + 21205962

0]Re 1198881119899 minus 1205960 ((31198891 + 1198892) 1198992ℓ

2minus 3119860) Im 1198881119899 + 119861 (1198891119899

2ℓ

2minus 2119860)Re 1198882119899 minus 1198611205960 Im 1198882119899

2119861 (12057223 + 1205722

4)

sdot 1205724

Re 120577 =(1198891119899

2ℓ

2minus 119860) (119862Re 1198881119899 minus 119860Re 1198882119899 minus 21205960 Im 1198882119899) + 1205960 (119862 Im 1198881119899 minus 119860 Im 1198882119899 + 21205960 Re 1198882119899)

2119861 (12057223 + 1205722

4)1205723

+(1198891119899

2ℓ2 minus 119860) (119862 Im 1198881119899 minus 119860 Im 1198882119899 + 21205960 Re 1198882119899) minus 1205960 (119862Re 1198881119899 minus 119860Re 1198882119899 minus 21205960 Im 1198882119899)

2119861 (12057223 + 1205722

4)1205724

Im 120577 =(1198891119899

2ℓ2 minus 119860) (119862 Im 1198881119899 minus 119860 Im 1198882119899 + 21205960 Re 1198882119899) minus 1205960 (119862Re 1198881119899 minus 119860Re 1198882119899 minus 21205960 Im 1198882119899)

2119861 (12057223 + 1205722

4)1205723

minus(1198891119899

2ℓ2 minus 119860) (119862Re 1198881119899 minus 119860Re 1198882119899 minus 21205960 Im 1198882119899) + 1205960 (119862 Im 1198881119899 minus 119860 Im 1198882119899 + 21205960 Re 1198882119899)2119861 (1205722

3 + 12057224)

1205724

Re 120578 =[(119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862] 1198891119899 + 119861 (51198891119899

2ℓ2 minus 2119860)1198892119899

21198611205725

Re 120578 =1205960 [(119860 + (31198892 minus 1198891) 119899

2ℓ2) 1198891119899 + 1198611198892119899]

21205725

Re 120574 =(1198891119899

2ℓ2 minus 119860) [1198621198891119899 + (411988911198992ℓ2 minus 119860) 1198892119899]

21198611205725

Im 120574 =1205960 [1198621198891119899 + (41198891119899

2ℓ2 minus 119860) 1198892119899]

21198611205725

Re120594 =[(119860 minus (1198891 + 1198892) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862] 1198891119899 + 119861 (1198891119899

2ℓ2 minus 2119860)1198892119899

21198611205726

Im120594 =1205960 [(119860 minus (1198891 + 1198892) 119899

2ℓ2) 1198891119899 + 1198611198892119899]

21205726119861 (1205820)

Re 120577 =(1198621198891119899 minus 1198601198892119899) (1198891119899

2ℓ

2minus 119860)

21198611205726

Im 120577 =1205960 (1198621198891119899 minus 1198601198892119899)

21198611205726

(74)

Journal of Applied Mathematics 11

Since ℓ120587119886lowast

119899= 1 minus 119894(119860 minus 1198891119899

2ℓ2)1205960 and ℓ120587119887lowast

119899= minus1198941198611205960

one can get

Re ⟨119902lowast 11987611990820119902

⟩ =14[1198911119906119906 (Re 120585 + 2 Re 120591)

+1198911119906V (Re 120578 + 2 Re120594) +1198911VV (Re 120574 + 2 Re 120577)]

+(119860 minus 1198891119899

2ℓ2)

41205960[1198911119906119906 (Im 120585 + 2 Im 120591)

+1198911119906V (Im 120578 + 2 Im120594) +1198911VV (Im 120574 + 2 Im 120577)]

+119861

41205960[1198912119906119906 (Im 120585 + 2 Im 120591)

+1198912119906V (Im 120578 + 2 Im120594) +1198912VV (Im 120574 + 2 Im 120577)]

Re ⟨119902lowast 11987611990811119902

⟩ =14[1198911119906119906 (120585 + 2120591)

+1198911119906V (Re 120578 + 2 Re120594) +1198911VV (Re 120574 + 2 Re 120577)]

+(119860 minus 1198891119899

2ℓ2)

41205960[1198911119906119906 (120585 + 2120591)

+1198911119906V (Re 120578 + 2 Re120594) +1198911VV (Re 120574 + 2 Re 120577)]

+119861

41205960[1198912119906V (Im 120578 + 2 Im120594)

+1198912VV (Im 120574 + 2 Im 120577)]

(75)

In addition it follows from intℓ120587

0 cos4(119899ℓ)119909 119889119909 = 3ℓ1205878and (62) that

⟨119902lowast 119862119902119902119902

⟩ =3ℓ1205878

(119886lowast

1198991198901119899 + 119887

lowast

1198991198902119899) (76)

Therefore

Re ⟨119902lowast 119862119902119902119902

⟩ =381198612 1198911119906119906119906119861

2

minus 31198911119906119906V (119860minus1198891119899

2

ℓ2)119861+ 21198911119906VV (119860minus

11988911198992

ℓ2)

2

+1198911VVV (119860minus1198891119899

2

ℓ2)119862+(119860minus

11988911198992

ℓ2)

sdot [1198911119906119906V119861minus 21198911119906VV (119860minus1198891119899

2

ℓ2)minus1198911VVV119862]

+119861[1198912119906119906V119861minus 21198912119906VV (119860minus1198891119899

2

ℓ2)minus1198912VVV119862]

(77)

Now by (40) we have

Re 1198881 (1205820) = Re ⟨119902lowast 11987611990811119902

⟩+12Re ⟨119902lowast 119876

11990820119902⟩+

12

sdotRe ⟨119902lowast 119862119902119902119902

⟩

=18[1198911119906119906 (Re 120585 + 2 Re 120591 + 2120585 + 4120591)

+1198911119906V (Re 120578 + 2 Re120594+ 2 Re 120578 + 4 Re120594)

+1198911VV (Re 120574 + 2 Re 120577 + 2 Re 120574 + 4 Re 120577)]

+(119860 minus 1198891119899

2ℓ2)

81205960[1198911119906119906 (Im 120585 + 2 Im 120591 + 2120585 + 4120591)

+1198911119906V (Im 120578 + 2 Im120594+ 2 Re 120578 + 4 Re120594)

+1198911VV (Im 120574 + 2 Im 120577 + 2 Re 120574 + 4 Re 120577)]

+119861

81205960[1198912119906119906 (Im 120585 + 2 Im 120591)

+1198912119906V (Im 120578119868+ 2 Im120594+ 2 Im 120578 + 4 Im120594)

+1198912VV (Im 120574 + Im 120577 + 2 Im 120574 + 4 Im 120577)]

+3

161198612 11989111199061199061199061198612minus 31198911119906119906V (119860minus

11988911198992

ℓ2)119861

+ 21198911119906VV (119860minus1198891119899

2

ℓ2)

2

+1198911VVV (119860minus1198891119899

2

ℓ2)119862

+(119860minus1198891119899

2

ℓ2)

sdot [1198911119906119906V119861minus 21198911119906VV (119860minus1198891119899

2

ℓ2)minus1198911VVV119862]

+119861[1198912119906119906V119861minus 21198912119906VV (119860minus1198891119899

2

ℓ2)minus1198912VVV119862]

(78)

Thus we have the following result

Theorem 5 Assume that condition (18) holds for 119899 isin N Thenthe spatially nonhomogeneous Hopf bifurcation of system (1) at(1205820 0 0) is supercritical (resp subcritical) if

Re 1198881 (1205820)

1198601015840 (1205820) + 1198631015840 (1205820)lt 0 (119903119890119904119901 gt 0) (79)

here Re 1198881(1205820) is given by (78)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

12 Journal of Applied Mathematics

Acknowledgments

Cun-Hua Zhang was supported by the Natural ScienceFoundation of Gansu Province (1212RJZA065) Xiang-PingYan was supported by the National Natural Science Foun-dation of China (11261028) Gansu Province Natural ScienceFoundation (145RJZA216) and China Scholarship Council

References

[1] A Andronov E A Leontovich I I Gordon and A G MaierTheory of Bifurcations of Dynamical Systems on a Plane IsraelProgram for Scientific Translations Jerusalem Israel 1971

[2] J Hale Theory of Functional Differential Equations SpringerNew York NY USA 1977

[3] B D Hassard N D Kazarinoff and Y H Wan Theory andApplications of Hopf Bifurcation Cambridge University PressCambridge UK 1981

[4] H Kielhofer Bifurcation Theory An Introduction with Appli-cations to PDEs vol 156 of Applied Mathematical SciencesSpringer New York NY USA 2004

[5] YA KuznetsovElements of Applied BifurcationTheory AppliedMathematical Sciences Springer New York NY USA 1998

[6] J-H Wu Theory and Applications of Partial Functional Differ-ential Equations Springer New York NY USA 1996

[7] F Yi JWei and J Shi ldquoBifurcation and spatiotemporal patternsin a homogeneous diffusive predator-prey systemrdquo Journal ofDifferential Equations vol 246 no 5 pp 1944ndash1977 2009

[8] L-J Yang and X-W Zeng ldquoStability of singular Hopf bifurca-tionsrdquo Journal of Differential Equations vol 206 no 1 pp 30ndash54 2004

[9] J Y Jin J P Shi J J Wei and F Q Yi ldquoBifurcations of patternedsolutions in the diffusive Lengyel-Epstein system of CIMAchemical reactionsrdquo Rocky Mountain Journal of Mathematicsvol 43 no 5 pp 1637ndash1674 2013

[10] S G Ruan ldquoDiffusion-driven instability in the Gierer-Mein-hardtmodel ofmorphogenesisrdquoNatural ResourceModeling vol11 no 2 pp 131ndash142 1998

[11] F Yi J Wei and J Shi ldquoDiffusion-driven instability andbifurcation in the Lengyel-Epstein systemrdquo Nonlinear AnalysisReal World Applications vol 9 no 3 pp 1038ndash1051 2008

[12] F Q Yi J J Wei and J P Shi ldquoGlobal asymptotical behavior ofthe Lengyel-Epstein reaction-diffusion systemrdquoApplied Mathe-matics Letters vol 22 no 1 pp 52ndash55 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 9: Research Article Normal Forms of Hopf Bifurcation for a ...

Journal of Applied Mathematics 9

119889119896119899

=119891119896119906119906

(119860 minus 11988911198992ℓ2) minus 2119891

119896119906V119861 minus 119891119896VV119862

119861

Re 119890119896119899

=119891119896119906119906119906

1198612 minus 3119891119896119906119906V (119860 minus 1198891119899

2ℓ2) 119861 + 2119891119896119906VV (119860 minus 1198891119899

2ℓ2)2+ 119891119896VVV (119860 minus 1198891119899

2ℓ2) 119862

1198612

Im 119890119896119899

=1205960 [119891119896119906119906V119861 minus 2119891

119896119906VV (119860 minus 11988911198992ℓ2) minus 119891

119896VVV119862]

1198612

119896 = 1 2(73)

Then from (70) and (73) we have

Re 120585 =[(119860 + (31198892 minus 1198891) 119899

2ℓ2)Re 1198881119899 + 119861Re 1198882119899 minus 21205960 Im 1198881119899] 1205721 + [(119860 + (31198892 minus 1198891) 119899

2ℓ2) Im 1198881119899 + 119861 Im 1198882119899 + 21205960 Re 1198881119899] 1205722

2 (12057221 + 1205722

2)

Im 120585 =[(119860 + (31198892 minus 1198891) 119899

2ℓ2) Im 1198881119899 + 119861 Im 1198882119899 + 21205960 Re 1198881119899] 1205721 minus [(119860 + (31198892 minus 1198891) 1198992ℓ2)Re 1198881119899 + 119861Re 1198882119899 minus 21205960 Im 1198881119899] 1205722

2 (12057221 + 1205722

2)

Re 120578

=[(119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0]Re 1198881119899 minus 31205960 ((1198891 minus 1198892) 1198992ℓ2 minus 119860) Im 1198881119899 + 119861 (51198891119899

2ℓ2 minus 2119860)Re 1198882119899 minus 1198611205960 Im 1198882119899

2119861 (12057221 + 1205722

2)

sdot 1205721

+[(119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0] Im 1198881119899 + 31205960 ((1198891 minus 1198892) 1198992ℓ2 minus 119860)Re 1198881119899 + 119861 (51198891119899

2ℓ2 minus 2119860) Im 1198882119899 + 1198611205960 Re 11988821198992119861 (1205722

1 + 12057222)

sdot 1205722

Im 120578

=[(119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0] Im 1198881119899 + 31205960 ((1198891 minus 1198892) 1198992ℓ2 minus 119860)Re 1198881119899 + 119861 (51198891119899

2ℓ2 minus 2119860) Im 1198882119899 + 1198611205960 Re 11988821198992119861 (1205722

1 + 12057222)

sdot 1205721

minus[(119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0]Re 1198881119899 minus 31205960 ((1198891 minus 1198892) 1198992ℓ2 minus 119860) Im 1198881119899 + 119861 (51198891119899

2ℓ2 minus 2119860)Re 1198882119899 minus 1198611205960 Im 1198882119899

2119861 (12057221 + 1205722

2)

sdot 1205722

Re 120574 =(1198891119899

2ℓ2 minus 119860) [119862Re 1198881119899 minus (119860 minus 411988911198992ℓ2)Re 1198882119899 minus 21205960 Im 1198882119899] + 1205960 [119862 Im 1198881119899 minus (119860 minus 41198891119899

2ℓ2) Im 1198882119899 + 21205960 Re 1198882119899]2119861 (1205722

1 + 12057222)

1205721

+(1198891119899

2ℓ2 minus 119860) [119862 Im 1198881119899 minus (119860 minus 411988911198992ℓ2) Im 1198882119899 + 21205960 Re 1198882119899] minus 1205960 [119862Re 1198881119899 minus (119860 minus 41198891119899

2ℓ2)Re 1198882119899 minus 21205960 Im 1198882119899]

2119861 (12057221 + 1205722

2)1205722

Im 120574 =(1198891119899

2ℓ2 minus 119860) [119862 Im 1198881119899 minus (119860 minus 411988911198992ℓ2) Im 1198882119899 + 21205960 Re 1198882119899] minus 1205960 [119862Re 1198881119899 minus (119860 minus 41198891119899

2ℓ2)Re 1198882119899 minus 21205960 Im 1198882119899]

2119861 (12057221 + 1205722

2)1205721

10 Journal of Applied Mathematics

minus(1198891119899

2ℓ2 minus 119860) [119862Re 1198881119899 minus (119860 minus 411988911198992ℓ2)Re 1198882119899 minus 21205960 Im 1198882119899] + 1205960 [119862 Im 1198881119899 minus (119860 minus 41198891119899

2ℓ2) Im 1198882119899 + 21205960 Re 1198882119899]2119861 (1205722

1 + 12057222)

1205722

Re 120591 =[(119860 + (1198891 + 1198892) 119899

2ℓ

2)Re 1198881119899 + 119861Re 1198882119899 minus 21205960 Im 1198881119899] 1205723 + [(119860 + (1198891 + 1198892) 119899

2ℓ

2) Im 1198881119899 + 119861 Im 1198882119899 + 21205960 Re 1198881119899] 1205724

2 (12057223 + 1205722

4)

Im 120591 =[(119860 + (1198891 + 1198892) 119899

2ℓ2) Im 1198881119899 + 119861 Im 1198882119899 + 21205960 Re 1198881119899] 1205723 minus [(119860 + (1198891 + 1198892) 1198992ℓ2)Re 1198881119899 + 119861Re 1198882119899 minus 21205960 Im 1198881119899] 1205724

2 (12057223 + 1205722

4)

Re120594

=[(119860 minus (1198891 + 1198892) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0]Re 1198881119899 minus 1205960 ((31198891 + 1198892) 1198992ℓ2 minus 3119860) Im 1198881119899 + 119861 (1198891119899

2ℓ2 minus 2119860)Re 1198882119899 minus 1198611205960 Im 1198882119899

2119861 (12057223 + 1205722

4)

sdot 1205723

+[(119860 minus (1198891 + 1198892) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0] Im 1198881119899 + 1205960 ((31198891 + 1198892) 1198992ℓ2 minus 3119860)Re 1198881119899 + 119861 (1198891119899

2ℓ2 minus 2119860) Im 1198882119899 + 1198611205960 Re 11988821198992119861 (1205722

3 + 12057224)

sdot 1205724

Im120594

=[(119860 minus (1198891 + 1198892) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0] Im 1198881119899 + 1205960 ((31198891 + 1198892) 1198992ℓ2 minus 3119860)Re 1198881119899 + 119861 (1198891119899

2ℓ2 minus 2119860) Im 1198882119899 + 1198611205960 Re 11988821198992119861 (1205722

3 + 12057224)

sdot 1205723

minus[(119860 minus (1198891 + 1198892) 119899

2ℓ

2) (1198891119899

2ℓ

2minus 119860) + 119861119862 + 21205962

0]Re 1198881119899 minus 1205960 ((31198891 + 1198892) 1198992ℓ

2minus 3119860) Im 1198881119899 + 119861 (1198891119899

2ℓ

2minus 2119860)Re 1198882119899 minus 1198611205960 Im 1198882119899

2119861 (12057223 + 1205722

4)

sdot 1205724

Re 120577 =(1198891119899

2ℓ

2minus 119860) (119862Re 1198881119899 minus 119860Re 1198882119899 minus 21205960 Im 1198882119899) + 1205960 (119862 Im 1198881119899 minus 119860 Im 1198882119899 + 21205960 Re 1198882119899)

2119861 (12057223 + 1205722

4)1205723

+(1198891119899

2ℓ2 minus 119860) (119862 Im 1198881119899 minus 119860 Im 1198882119899 + 21205960 Re 1198882119899) minus 1205960 (119862Re 1198881119899 minus 119860Re 1198882119899 minus 21205960 Im 1198882119899)

2119861 (12057223 + 1205722

4)1205724

Im 120577 =(1198891119899

2ℓ2 minus 119860) (119862 Im 1198881119899 minus 119860 Im 1198882119899 + 21205960 Re 1198882119899) minus 1205960 (119862Re 1198881119899 minus 119860Re 1198882119899 minus 21205960 Im 1198882119899)

2119861 (12057223 + 1205722

4)1205723

minus(1198891119899

2ℓ2 minus 119860) (119862Re 1198881119899 minus 119860Re 1198882119899 minus 21205960 Im 1198882119899) + 1205960 (119862 Im 1198881119899 minus 119860 Im 1198882119899 + 21205960 Re 1198882119899)2119861 (1205722

3 + 12057224)

1205724

Re 120578 =[(119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862] 1198891119899 + 119861 (51198891119899

2ℓ2 minus 2119860)1198892119899

21198611205725

Re 120578 =1205960 [(119860 + (31198892 minus 1198891) 119899

2ℓ2) 1198891119899 + 1198611198892119899]

21205725

Re 120574 =(1198891119899

2ℓ2 minus 119860) [1198621198891119899 + (411988911198992ℓ2 minus 119860) 1198892119899]

21198611205725

Im 120574 =1205960 [1198621198891119899 + (41198891119899

2ℓ2 minus 119860) 1198892119899]

21198611205725

Re120594 =[(119860 minus (1198891 + 1198892) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862] 1198891119899 + 119861 (1198891119899

2ℓ2 minus 2119860)1198892119899

21198611205726

Im120594 =1205960 [(119860 minus (1198891 + 1198892) 119899

2ℓ2) 1198891119899 + 1198611198892119899]

21205726119861 (1205820)

Re 120577 =(1198621198891119899 minus 1198601198892119899) (1198891119899

2ℓ

2minus 119860)

21198611205726

Im 120577 =1205960 (1198621198891119899 minus 1198601198892119899)

21198611205726

(74)

Journal of Applied Mathematics 11

Since ℓ120587119886lowast

119899= 1 minus 119894(119860 minus 1198891119899

2ℓ2)1205960 and ℓ120587119887lowast

119899= minus1198941198611205960

one can get

Re ⟨119902lowast 11987611990820119902

⟩ =14[1198911119906119906 (Re 120585 + 2 Re 120591)

+1198911119906V (Re 120578 + 2 Re120594) +1198911VV (Re 120574 + 2 Re 120577)]

+(119860 minus 1198891119899

2ℓ2)

41205960[1198911119906119906 (Im 120585 + 2 Im 120591)

+1198911119906V (Im 120578 + 2 Im120594) +1198911VV (Im 120574 + 2 Im 120577)]

+119861

41205960[1198912119906119906 (Im 120585 + 2 Im 120591)

+1198912119906V (Im 120578 + 2 Im120594) +1198912VV (Im 120574 + 2 Im 120577)]

Re ⟨119902lowast 11987611990811119902

⟩ =14[1198911119906119906 (120585 + 2120591)

+1198911119906V (Re 120578 + 2 Re120594) +1198911VV (Re 120574 + 2 Re 120577)]

+(119860 minus 1198891119899

2ℓ2)

41205960[1198911119906119906 (120585 + 2120591)

+1198911119906V (Re 120578 + 2 Re120594) +1198911VV (Re 120574 + 2 Re 120577)]

+119861

41205960[1198912119906V (Im 120578 + 2 Im120594)

+1198912VV (Im 120574 + 2 Im 120577)]

(75)

In addition it follows from intℓ120587

0 cos4(119899ℓ)119909 119889119909 = 3ℓ1205878and (62) that

⟨119902lowast 119862119902119902119902

⟩ =3ℓ1205878

(119886lowast

1198991198901119899 + 119887

lowast

1198991198902119899) (76)

Therefore

Re ⟨119902lowast 119862119902119902119902

⟩ =381198612 1198911119906119906119906119861

2

minus 31198911119906119906V (119860minus1198891119899

2

ℓ2)119861+ 21198911119906VV (119860minus

11988911198992

ℓ2)

2

+1198911VVV (119860minus1198891119899

2

ℓ2)119862+(119860minus

11988911198992

ℓ2)

sdot [1198911119906119906V119861minus 21198911119906VV (119860minus1198891119899

2

ℓ2)minus1198911VVV119862]

+119861[1198912119906119906V119861minus 21198912119906VV (119860minus1198891119899

2

ℓ2)minus1198912VVV119862]

(77)

Now by (40) we have

Re 1198881 (1205820) = Re ⟨119902lowast 11987611990811119902

⟩+12Re ⟨119902lowast 119876

11990820119902⟩+

12

sdotRe ⟨119902lowast 119862119902119902119902

⟩

=18[1198911119906119906 (Re 120585 + 2 Re 120591 + 2120585 + 4120591)

+1198911119906V (Re 120578 + 2 Re120594+ 2 Re 120578 + 4 Re120594)

+1198911VV (Re 120574 + 2 Re 120577 + 2 Re 120574 + 4 Re 120577)]

+(119860 minus 1198891119899

2ℓ2)

81205960[1198911119906119906 (Im 120585 + 2 Im 120591 + 2120585 + 4120591)

+1198911119906V (Im 120578 + 2 Im120594+ 2 Re 120578 + 4 Re120594)

+1198911VV (Im 120574 + 2 Im 120577 + 2 Re 120574 + 4 Re 120577)]

+119861

81205960[1198912119906119906 (Im 120585 + 2 Im 120591)

+1198912119906V (Im 120578119868+ 2 Im120594+ 2 Im 120578 + 4 Im120594)

+1198912VV (Im 120574 + Im 120577 + 2 Im 120574 + 4 Im 120577)]

+3

161198612 11989111199061199061199061198612minus 31198911119906119906V (119860minus

11988911198992

ℓ2)119861

+ 21198911119906VV (119860minus1198891119899

2

ℓ2)

2

+1198911VVV (119860minus1198891119899

2

ℓ2)119862

+(119860minus1198891119899

2

ℓ2)

sdot [1198911119906119906V119861minus 21198911119906VV (119860minus1198891119899

2

ℓ2)minus1198911VVV119862]

+119861[1198912119906119906V119861minus 21198912119906VV (119860minus1198891119899

2

ℓ2)minus1198912VVV119862]

(78)

Thus we have the following result

Theorem 5 Assume that condition (18) holds for 119899 isin N Thenthe spatially nonhomogeneous Hopf bifurcation of system (1) at(1205820 0 0) is supercritical (resp subcritical) if

Re 1198881 (1205820)

1198601015840 (1205820) + 1198631015840 (1205820)lt 0 (119903119890119904119901 gt 0) (79)

here Re 1198881(1205820) is given by (78)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

12 Journal of Applied Mathematics

Acknowledgments

Cun-Hua Zhang was supported by the Natural ScienceFoundation of Gansu Province (1212RJZA065) Xiang-PingYan was supported by the National Natural Science Foun-dation of China (11261028) Gansu Province Natural ScienceFoundation (145RJZA216) and China Scholarship Council

References

[1] A Andronov E A Leontovich I I Gordon and A G MaierTheory of Bifurcations of Dynamical Systems on a Plane IsraelProgram for Scientific Translations Jerusalem Israel 1971

[2] J Hale Theory of Functional Differential Equations SpringerNew York NY USA 1977

[3] B D Hassard N D Kazarinoff and Y H Wan Theory andApplications of Hopf Bifurcation Cambridge University PressCambridge UK 1981

[4] H Kielhofer Bifurcation Theory An Introduction with Appli-cations to PDEs vol 156 of Applied Mathematical SciencesSpringer New York NY USA 2004

[5] YA KuznetsovElements of Applied BifurcationTheory AppliedMathematical Sciences Springer New York NY USA 1998

[6] J-H Wu Theory and Applications of Partial Functional Differ-ential Equations Springer New York NY USA 1996

[7] F Yi JWei and J Shi ldquoBifurcation and spatiotemporal patternsin a homogeneous diffusive predator-prey systemrdquo Journal ofDifferential Equations vol 246 no 5 pp 1944ndash1977 2009

[8] L-J Yang and X-W Zeng ldquoStability of singular Hopf bifurca-tionsrdquo Journal of Differential Equations vol 206 no 1 pp 30ndash54 2004

[9] J Y Jin J P Shi J J Wei and F Q Yi ldquoBifurcations of patternedsolutions in the diffusive Lengyel-Epstein system of CIMAchemical reactionsrdquo Rocky Mountain Journal of Mathematicsvol 43 no 5 pp 1637ndash1674 2013

[10] S G Ruan ldquoDiffusion-driven instability in the Gierer-Mein-hardtmodel ofmorphogenesisrdquoNatural ResourceModeling vol11 no 2 pp 131ndash142 1998

[11] F Yi J Wei and J Shi ldquoDiffusion-driven instability andbifurcation in the Lengyel-Epstein systemrdquo Nonlinear AnalysisReal World Applications vol 9 no 3 pp 1038ndash1051 2008

[12] F Q Yi J J Wei and J P Shi ldquoGlobal asymptotical behavior ofthe Lengyel-Epstein reaction-diffusion systemrdquoApplied Mathe-matics Letters vol 22 no 1 pp 52ndash55 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 10: Research Article Normal Forms of Hopf Bifurcation for a ...

10 Journal of Applied Mathematics

minus(1198891119899

2ℓ2 minus 119860) [119862Re 1198881119899 minus (119860 minus 411988911198992ℓ2)Re 1198882119899 minus 21205960 Im 1198882119899] + 1205960 [119862 Im 1198881119899 minus (119860 minus 41198891119899

2ℓ2) Im 1198882119899 + 21205960 Re 1198882119899]2119861 (1205722

1 + 12057222)

1205722

Re 120591 =[(119860 + (1198891 + 1198892) 119899

2ℓ

2)Re 1198881119899 + 119861Re 1198882119899 minus 21205960 Im 1198881119899] 1205723 + [(119860 + (1198891 + 1198892) 119899

2ℓ

2) Im 1198881119899 + 119861 Im 1198882119899 + 21205960 Re 1198881119899] 1205724

2 (12057223 + 1205722

4)

Im 120591 =[(119860 + (1198891 + 1198892) 119899

2ℓ2) Im 1198881119899 + 119861 Im 1198882119899 + 21205960 Re 1198881119899] 1205723 minus [(119860 + (1198891 + 1198892) 1198992ℓ2)Re 1198881119899 + 119861Re 1198882119899 minus 21205960 Im 1198881119899] 1205724

2 (12057223 + 1205722

4)

Re120594

=[(119860 minus (1198891 + 1198892) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0]Re 1198881119899 minus 1205960 ((31198891 + 1198892) 1198992ℓ2 minus 3119860) Im 1198881119899 + 119861 (1198891119899

2ℓ2 minus 2119860)Re 1198882119899 minus 1198611205960 Im 1198882119899

2119861 (12057223 + 1205722

4)

sdot 1205723

+[(119860 minus (1198891 + 1198892) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0] Im 1198881119899 + 1205960 ((31198891 + 1198892) 1198992ℓ2 minus 3119860)Re 1198881119899 + 119861 (1198891119899

2ℓ2 minus 2119860) Im 1198882119899 + 1198611205960 Re 11988821198992119861 (1205722

3 + 12057224)

sdot 1205724

Im120594

=[(119860 minus (1198891 + 1198892) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862 + 21205962

0] Im 1198881119899 + 1205960 ((31198891 + 1198892) 1198992ℓ2 minus 3119860)Re 1198881119899 + 119861 (1198891119899

2ℓ2 minus 2119860) Im 1198882119899 + 1198611205960 Re 11988821198992119861 (1205722

3 + 12057224)

sdot 1205723

minus[(119860 minus (1198891 + 1198892) 119899

2ℓ

2) (1198891119899

2ℓ

2minus 119860) + 119861119862 + 21205962

0]Re 1198881119899 minus 1205960 ((31198891 + 1198892) 1198992ℓ

2minus 3119860) Im 1198881119899 + 119861 (1198891119899

2ℓ

2minus 2119860)Re 1198882119899 minus 1198611205960 Im 1198882119899

2119861 (12057223 + 1205722

4)

sdot 1205724

Re 120577 =(1198891119899

2ℓ

2minus 119860) (119862Re 1198881119899 minus 119860Re 1198882119899 minus 21205960 Im 1198882119899) + 1205960 (119862 Im 1198881119899 minus 119860 Im 1198882119899 + 21205960 Re 1198882119899)

2119861 (12057223 + 1205722

4)1205723

+(1198891119899

2ℓ2 minus 119860) (119862 Im 1198881119899 minus 119860 Im 1198882119899 + 21205960 Re 1198882119899) minus 1205960 (119862Re 1198881119899 minus 119860Re 1198882119899 minus 21205960 Im 1198882119899)

2119861 (12057223 + 1205722

4)1205724

Im 120577 =(1198891119899

2ℓ2 minus 119860) (119862 Im 1198881119899 minus 119860 Im 1198882119899 + 21205960 Re 1198882119899) minus 1205960 (119862Re 1198881119899 minus 119860Re 1198882119899 minus 21205960 Im 1198882119899)

2119861 (12057223 + 1205722

4)1205723

minus(1198891119899

2ℓ2 minus 119860) (119862Re 1198881119899 minus 119860Re 1198882119899 minus 21205960 Im 1198882119899) + 1205960 (119862 Im 1198881119899 minus 119860 Im 1198882119899 + 21205960 Re 1198882119899)2119861 (1205722

3 + 12057224)

1205724

Re 120578 =[(119860 + (31198892 minus 1198891) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862] 1198891119899 + 119861 (51198891119899

2ℓ2 minus 2119860)1198892119899

21198611205725

Re 120578 =1205960 [(119860 + (31198892 minus 1198891) 119899

2ℓ2) 1198891119899 + 1198611198892119899]

21205725

Re 120574 =(1198891119899

2ℓ2 minus 119860) [1198621198891119899 + (411988911198992ℓ2 minus 119860) 1198892119899]

21198611205725

Im 120574 =1205960 [1198621198891119899 + (41198891119899

2ℓ2 minus 119860) 1198892119899]

21198611205725

Re120594 =[(119860 minus (1198891 + 1198892) 119899

2ℓ2) (11988911198992ℓ2 minus 119860) + 119861119862] 1198891119899 + 119861 (1198891119899

2ℓ2 minus 2119860)1198892119899

21198611205726

Im120594 =1205960 [(119860 minus (1198891 + 1198892) 119899

2ℓ2) 1198891119899 + 1198611198892119899]

21205726119861 (1205820)

Re 120577 =(1198621198891119899 minus 1198601198892119899) (1198891119899

2ℓ

2minus 119860)

21198611205726

Im 120577 =1205960 (1198621198891119899 minus 1198601198892119899)

21198611205726

(74)

Journal of Applied Mathematics 11

Since ℓ120587119886lowast

119899= 1 minus 119894(119860 minus 1198891119899

2ℓ2)1205960 and ℓ120587119887lowast

119899= minus1198941198611205960

one can get

Re ⟨119902lowast 11987611990820119902

⟩ =14[1198911119906119906 (Re 120585 + 2 Re 120591)

+1198911119906V (Re 120578 + 2 Re120594) +1198911VV (Re 120574 + 2 Re 120577)]

+(119860 minus 1198891119899

2ℓ2)

41205960[1198911119906119906 (Im 120585 + 2 Im 120591)

+1198911119906V (Im 120578 + 2 Im120594) +1198911VV (Im 120574 + 2 Im 120577)]

+119861

41205960[1198912119906119906 (Im 120585 + 2 Im 120591)

+1198912119906V (Im 120578 + 2 Im120594) +1198912VV (Im 120574 + 2 Im 120577)]

Re ⟨119902lowast 11987611990811119902

⟩ =14[1198911119906119906 (120585 + 2120591)

+1198911119906V (Re 120578 + 2 Re120594) +1198911VV (Re 120574 + 2 Re 120577)]

+(119860 minus 1198891119899

2ℓ2)

41205960[1198911119906119906 (120585 + 2120591)

+1198911119906V (Re 120578 + 2 Re120594) +1198911VV (Re 120574 + 2 Re 120577)]

+119861

41205960[1198912119906V (Im 120578 + 2 Im120594)

+1198912VV (Im 120574 + 2 Im 120577)]

(75)

In addition it follows from intℓ120587

0 cos4(119899ℓ)119909 119889119909 = 3ℓ1205878and (62) that

⟨119902lowast 119862119902119902119902

⟩ =3ℓ1205878

(119886lowast

1198991198901119899 + 119887

lowast

1198991198902119899) (76)

Therefore

Re ⟨119902lowast 119862119902119902119902

⟩ =381198612 1198911119906119906119906119861

2

minus 31198911119906119906V (119860minus1198891119899

2

ℓ2)119861+ 21198911119906VV (119860minus

11988911198992

ℓ2)

2

+1198911VVV (119860minus1198891119899

2

ℓ2)119862+(119860minus

11988911198992

ℓ2)

sdot [1198911119906119906V119861minus 21198911119906VV (119860minus1198891119899

2

ℓ2)minus1198911VVV119862]

+119861[1198912119906119906V119861minus 21198912119906VV (119860minus1198891119899

2

ℓ2)minus1198912VVV119862]

(77)

Now by (40) we have

Re 1198881 (1205820) = Re ⟨119902lowast 11987611990811119902

⟩+12Re ⟨119902lowast 119876

11990820119902⟩+

12

sdotRe ⟨119902lowast 119862119902119902119902

⟩

=18[1198911119906119906 (Re 120585 + 2 Re 120591 + 2120585 + 4120591)

+1198911119906V (Re 120578 + 2 Re120594+ 2 Re 120578 + 4 Re120594)

+1198911VV (Re 120574 + 2 Re 120577 + 2 Re 120574 + 4 Re 120577)]

+(119860 minus 1198891119899

2ℓ2)

81205960[1198911119906119906 (Im 120585 + 2 Im 120591 + 2120585 + 4120591)

+1198911119906V (Im 120578 + 2 Im120594+ 2 Re 120578 + 4 Re120594)

+1198911VV (Im 120574 + 2 Im 120577 + 2 Re 120574 + 4 Re 120577)]

+119861

81205960[1198912119906119906 (Im 120585 + 2 Im 120591)

+1198912119906V (Im 120578119868+ 2 Im120594+ 2 Im 120578 + 4 Im120594)

+1198912VV (Im 120574 + Im 120577 + 2 Im 120574 + 4 Im 120577)]

+3

161198612 11989111199061199061199061198612minus 31198911119906119906V (119860minus

11988911198992

ℓ2)119861

+ 21198911119906VV (119860minus1198891119899

2

ℓ2)

2

+1198911VVV (119860minus1198891119899

2

ℓ2)119862

+(119860minus1198891119899

2

ℓ2)

sdot [1198911119906119906V119861minus 21198911119906VV (119860minus1198891119899

2

ℓ2)minus1198911VVV119862]

+119861[1198912119906119906V119861minus 21198912119906VV (119860minus1198891119899

2

ℓ2)minus1198912VVV119862]

(78)

Thus we have the following result

Theorem 5 Assume that condition (18) holds for 119899 isin N Thenthe spatially nonhomogeneous Hopf bifurcation of system (1) at(1205820 0 0) is supercritical (resp subcritical) if

Re 1198881 (1205820)

1198601015840 (1205820) + 1198631015840 (1205820)lt 0 (119903119890119904119901 gt 0) (79)

here Re 1198881(1205820) is given by (78)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

12 Journal of Applied Mathematics

Acknowledgments

Cun-Hua Zhang was supported by the Natural ScienceFoundation of Gansu Province (1212RJZA065) Xiang-PingYan was supported by the National Natural Science Foun-dation of China (11261028) Gansu Province Natural ScienceFoundation (145RJZA216) and China Scholarship Council

References

[1] A Andronov E A Leontovich I I Gordon and A G MaierTheory of Bifurcations of Dynamical Systems on a Plane IsraelProgram for Scientific Translations Jerusalem Israel 1971

[2] J Hale Theory of Functional Differential Equations SpringerNew York NY USA 1977

[3] B D Hassard N D Kazarinoff and Y H Wan Theory andApplications of Hopf Bifurcation Cambridge University PressCambridge UK 1981

[4] H Kielhofer Bifurcation Theory An Introduction with Appli-cations to PDEs vol 156 of Applied Mathematical SciencesSpringer New York NY USA 2004

[5] YA KuznetsovElements of Applied BifurcationTheory AppliedMathematical Sciences Springer New York NY USA 1998

[6] J-H Wu Theory and Applications of Partial Functional Differ-ential Equations Springer New York NY USA 1996

[7] F Yi JWei and J Shi ldquoBifurcation and spatiotemporal patternsin a homogeneous diffusive predator-prey systemrdquo Journal ofDifferential Equations vol 246 no 5 pp 1944ndash1977 2009

[8] L-J Yang and X-W Zeng ldquoStability of singular Hopf bifurca-tionsrdquo Journal of Differential Equations vol 206 no 1 pp 30ndash54 2004

[9] J Y Jin J P Shi J J Wei and F Q Yi ldquoBifurcations of patternedsolutions in the diffusive Lengyel-Epstein system of CIMAchemical reactionsrdquo Rocky Mountain Journal of Mathematicsvol 43 no 5 pp 1637ndash1674 2013

[10] S G Ruan ldquoDiffusion-driven instability in the Gierer-Mein-hardtmodel ofmorphogenesisrdquoNatural ResourceModeling vol11 no 2 pp 131ndash142 1998

[11] F Yi J Wei and J Shi ldquoDiffusion-driven instability andbifurcation in the Lengyel-Epstein systemrdquo Nonlinear AnalysisReal World Applications vol 9 no 3 pp 1038ndash1051 2008

[12] F Q Yi J J Wei and J P Shi ldquoGlobal asymptotical behavior ofthe Lengyel-Epstein reaction-diffusion systemrdquoApplied Mathe-matics Letters vol 22 no 1 pp 52ndash55 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 11: Research Article Normal Forms of Hopf Bifurcation for a ...

Journal of Applied Mathematics 11

Since ℓ120587119886lowast

119899= 1 minus 119894(119860 minus 1198891119899

2ℓ2)1205960 and ℓ120587119887lowast

119899= minus1198941198611205960

one can get

Re ⟨119902lowast 11987611990820119902

⟩ =14[1198911119906119906 (Re 120585 + 2 Re 120591)

+1198911119906V (Re 120578 + 2 Re120594) +1198911VV (Re 120574 + 2 Re 120577)]

+(119860 minus 1198891119899

2ℓ2)

41205960[1198911119906119906 (Im 120585 + 2 Im 120591)

+1198911119906V (Im 120578 + 2 Im120594) +1198911VV (Im 120574 + 2 Im 120577)]

+119861

41205960[1198912119906119906 (Im 120585 + 2 Im 120591)

+1198912119906V (Im 120578 + 2 Im120594) +1198912VV (Im 120574 + 2 Im 120577)]

Re ⟨119902lowast 11987611990811119902

⟩ =14[1198911119906119906 (120585 + 2120591)

+1198911119906V (Re 120578 + 2 Re120594) +1198911VV (Re 120574 + 2 Re 120577)]

+(119860 minus 1198891119899

2ℓ2)

41205960[1198911119906119906 (120585 + 2120591)

+1198911119906V (Re 120578 + 2 Re120594) +1198911VV (Re 120574 + 2 Re 120577)]

+119861

41205960[1198912119906V (Im 120578 + 2 Im120594)

+1198912VV (Im 120574 + 2 Im 120577)]

(75)

In addition it follows from intℓ120587

0 cos4(119899ℓ)119909 119889119909 = 3ℓ1205878and (62) that

⟨119902lowast 119862119902119902119902

⟩ =3ℓ1205878

(119886lowast

1198991198901119899 + 119887

lowast

1198991198902119899) (76)

Therefore

Re ⟨119902lowast 119862119902119902119902

⟩ =381198612 1198911119906119906119906119861

2

minus 31198911119906119906V (119860minus1198891119899

2

ℓ2)119861+ 21198911119906VV (119860minus

11988911198992

ℓ2)

2

+1198911VVV (119860minus1198891119899

2

ℓ2)119862+(119860minus

11988911198992

ℓ2)

sdot [1198911119906119906V119861minus 21198911119906VV (119860minus1198891119899

2

ℓ2)minus1198911VVV119862]

+119861[1198912119906119906V119861minus 21198912119906VV (119860minus1198891119899

2

ℓ2)minus1198912VVV119862]

(77)

Now by (40) we have

Re 1198881 (1205820) = Re ⟨119902lowast 11987611990811119902

⟩+12Re ⟨119902lowast 119876

11990820119902⟩+

12

sdotRe ⟨119902lowast 119862119902119902119902

⟩

=18[1198911119906119906 (Re 120585 + 2 Re 120591 + 2120585 + 4120591)

+1198911119906V (Re 120578 + 2 Re120594+ 2 Re 120578 + 4 Re120594)

+1198911VV (Re 120574 + 2 Re 120577 + 2 Re 120574 + 4 Re 120577)]

+(119860 minus 1198891119899

2ℓ2)

81205960[1198911119906119906 (Im 120585 + 2 Im 120591 + 2120585 + 4120591)

+1198911119906V (Im 120578 + 2 Im120594+ 2 Re 120578 + 4 Re120594)

+1198911VV (Im 120574 + 2 Im 120577 + 2 Re 120574 + 4 Re 120577)]

+119861

81205960[1198912119906119906 (Im 120585 + 2 Im 120591)

+1198912119906V (Im 120578119868+ 2 Im120594+ 2 Im 120578 + 4 Im120594)

+1198912VV (Im 120574 + Im 120577 + 2 Im 120574 + 4 Im 120577)]

+3

161198612 11989111199061199061199061198612minus 31198911119906119906V (119860minus

11988911198992

ℓ2)119861

+ 21198911119906VV (119860minus1198891119899

2

ℓ2)

2

+1198911VVV (119860minus1198891119899

2

ℓ2)119862

+(119860minus1198891119899

2

ℓ2)

sdot [1198911119906119906V119861minus 21198911119906VV (119860minus1198891119899

2

ℓ2)minus1198911VVV119862]

+119861[1198912119906119906V119861minus 21198912119906VV (119860minus1198891119899

2

ℓ2)minus1198912VVV119862]

(78)

Thus we have the following result

Theorem 5 Assume that condition (18) holds for 119899 isin N Thenthe spatially nonhomogeneous Hopf bifurcation of system (1) at(1205820 0 0) is supercritical (resp subcritical) if

Re 1198881 (1205820)

1198601015840 (1205820) + 1198631015840 (1205820)lt 0 (119903119890119904119901 gt 0) (79)

here Re 1198881(1205820) is given by (78)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

12 Journal of Applied Mathematics

Acknowledgments

Cun-Hua Zhang was supported by the Natural ScienceFoundation of Gansu Province (1212RJZA065) Xiang-PingYan was supported by the National Natural Science Foun-dation of China (11261028) Gansu Province Natural ScienceFoundation (145RJZA216) and China Scholarship Council

References

[1] A Andronov E A Leontovich I I Gordon and A G MaierTheory of Bifurcations of Dynamical Systems on a Plane IsraelProgram for Scientific Translations Jerusalem Israel 1971

[2] J Hale Theory of Functional Differential Equations SpringerNew York NY USA 1977

[3] B D Hassard N D Kazarinoff and Y H Wan Theory andApplications of Hopf Bifurcation Cambridge University PressCambridge UK 1981

[4] H Kielhofer Bifurcation Theory An Introduction with Appli-cations to PDEs vol 156 of Applied Mathematical SciencesSpringer New York NY USA 2004

[5] YA KuznetsovElements of Applied BifurcationTheory AppliedMathematical Sciences Springer New York NY USA 1998

[6] J-H Wu Theory and Applications of Partial Functional Differ-ential Equations Springer New York NY USA 1996

[7] F Yi JWei and J Shi ldquoBifurcation and spatiotemporal patternsin a homogeneous diffusive predator-prey systemrdquo Journal ofDifferential Equations vol 246 no 5 pp 1944ndash1977 2009

[8] L-J Yang and X-W Zeng ldquoStability of singular Hopf bifurca-tionsrdquo Journal of Differential Equations vol 206 no 1 pp 30ndash54 2004

[9] J Y Jin J P Shi J J Wei and F Q Yi ldquoBifurcations of patternedsolutions in the diffusive Lengyel-Epstein system of CIMAchemical reactionsrdquo Rocky Mountain Journal of Mathematicsvol 43 no 5 pp 1637ndash1674 2013

[10] S G Ruan ldquoDiffusion-driven instability in the Gierer-Mein-hardtmodel ofmorphogenesisrdquoNatural ResourceModeling vol11 no 2 pp 131ndash142 1998

[11] F Yi J Wei and J Shi ldquoDiffusion-driven instability andbifurcation in the Lengyel-Epstein systemrdquo Nonlinear AnalysisReal World Applications vol 9 no 3 pp 1038ndash1051 2008

[12] F Q Yi J J Wei and J P Shi ldquoGlobal asymptotical behavior ofthe Lengyel-Epstein reaction-diffusion systemrdquoApplied Mathe-matics Letters vol 22 no 1 pp 52ndash55 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 12: Research Article Normal Forms of Hopf Bifurcation for a ...

12 Journal of Applied Mathematics

Acknowledgments

Cun-Hua Zhang was supported by the Natural ScienceFoundation of Gansu Province (1212RJZA065) Xiang-PingYan was supported by the National Natural Science Foun-dation of China (11261028) Gansu Province Natural ScienceFoundation (145RJZA216) and China Scholarship Council

References

[1] A Andronov E A Leontovich I I Gordon and A G MaierTheory of Bifurcations of Dynamical Systems on a Plane IsraelProgram for Scientific Translations Jerusalem Israel 1971

[2] J Hale Theory of Functional Differential Equations SpringerNew York NY USA 1977

[3] B D Hassard N D Kazarinoff and Y H Wan Theory andApplications of Hopf Bifurcation Cambridge University PressCambridge UK 1981

[4] H Kielhofer Bifurcation Theory An Introduction with Appli-cations to PDEs vol 156 of Applied Mathematical SciencesSpringer New York NY USA 2004

[5] YA KuznetsovElements of Applied BifurcationTheory AppliedMathematical Sciences Springer New York NY USA 1998

[6] J-H Wu Theory and Applications of Partial Functional Differ-ential Equations Springer New York NY USA 1996

[7] F Yi JWei and J Shi ldquoBifurcation and spatiotemporal patternsin a homogeneous diffusive predator-prey systemrdquo Journal ofDifferential Equations vol 246 no 5 pp 1944ndash1977 2009

[8] L-J Yang and X-W Zeng ldquoStability of singular Hopf bifurca-tionsrdquo Journal of Differential Equations vol 206 no 1 pp 30ndash54 2004

[9] J Y Jin J P Shi J J Wei and F Q Yi ldquoBifurcations of patternedsolutions in the diffusive Lengyel-Epstein system of CIMAchemical reactionsrdquo Rocky Mountain Journal of Mathematicsvol 43 no 5 pp 1637ndash1674 2013

[10] S G Ruan ldquoDiffusion-driven instability in the Gierer-Mein-hardtmodel ofmorphogenesisrdquoNatural ResourceModeling vol11 no 2 pp 131ndash142 1998

[11] F Yi J Wei and J Shi ldquoDiffusion-driven instability andbifurcation in the Lengyel-Epstein systemrdquo Nonlinear AnalysisReal World Applications vol 9 no 3 pp 1038ndash1051 2008

[12] F Q Yi J J Wei and J P Shi ldquoGlobal asymptotical behavior ofthe Lengyel-Epstein reaction-diffusion systemrdquoApplied Mathe-matics Letters vol 22 no 1 pp 52ndash55 2009

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 13: Research Article Normal Forms of Hopf Bifurcation for a ...

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Generic Hopf bifurcation from lines of equilibria without parameters: I. Theorydynamics.mi.fu-berlin.de/.../FieLieAle98-HopfTheory.pdf · 2002-08-21 · Generic Hopf bifurcation from

Generic Hopf bifurcation from lines of equilibria without parameters: I. Theorydynamics.mi.fu-berlin.de/.../FieLieAle98-HopfTheory.pdf · 2002-08-21 · Generic Hopf bifurcation from

Hopf Bifurcation in Coupled Cell Networks with Interior Symmetries · 2017. 12. 18. · Hopf Bifurcation in Coupled Cell Networks with Interior Symmetries Fernando Antoneli1 Ana Paula

Hopf Bifurcation in Coupled Cell Networks with Interior Symmetries · 2017. 12. 18. · Hopf Bifurcation in Coupled Cell Networks with Interior Symmetries Fernando Antoneli1 Ana Paula

The Boundary-Hopf-Fold Bifurcation in Filippov SystemsThe Boundary-Hopf-Fold Bifurcation in Filippov Systems ... Such systems find applications in several fields, for example, mechanical

The Boundary-Hopf-Fold Bifurcation in Filippov SystemsThe Boundary-Hopf-Fold Bifurcation in Filippov Systems ... Such systems find applications in several fields, for example, mechanical

Research Article Hopf-Pitchfork Bifurcation in a ...the system ( ) undergoes a Pitchfork bifurcation at the positive equilibrium. 3. Hopf-Pitchfork Bifurcation Inthefollowing,weconsiderthecasethat(

Research Article Hopf-Pitchfork Bifurcation in a ...the system ( ) undergoes a Pitchfork bifurcation at the positive equilibrium. 3. Hopf-Pitchfork Bifurcation Inthefollowing,weconsiderthecasethat(

Introduction to Bifurcations and The Hopf Bifurcation ...shipman/47/volume3b2011/M640_MunozAlicea.pdfIntroduction to Bifurcations and The Hopf Bifurcation Theorem Roberto Munoz-Alicea~-3

Introduction to Bifurcations and The Hopf Bifurcation ...shipman/47/volume3b2011/M640_MunozAlicea.pdfIntroduction to Bifurcations and The Hopf Bifurcation Theorem Roberto Munoz-Alicea~-3

Asymmetry and Hopf bifurcation in spherical …Asymmetry and Hopf bifurcation in spherical Couette flow Chowdhury K. Mamun Dtprrtnwnr of Chemical l3~girtccring, Chivcrsity of l2m.1.~

Asymmetry and Hopf bifurcation in spherical …Asymmetry and Hopf bifurcation in spherical Couette flow Chowdhury K. Mamun Dtprrtnwnr of Chemical l3~girtccring, Chivcrsity of l2m.1.~

Hopf Bifurcation and Chaotic Response in Nonlinear ...Hopf Bifurcation and Chaotic Response in Nonlinear ... rate recurrent neural networks are used to study the neuronal behaviour

Hopf Bifurcation and Chaotic Response in Nonlinear ...Hopf Bifurcation and Chaotic Response in Nonlinear ... rate recurrent neural networks are used to study the neuronal behaviour

Bifurcation analysis of chemical reaction mechanisms. II. Hopf bifurcation analysishopf.chem.brandeis.edu/pubs/pub208 rep.pdf · Bifurcation analysis of chemical reaction mechanisms.

Bifurcation analysis of chemical reaction mechanisms. II. Hopf bifurcation analysishopf.chem.brandeis.edu/pubs/pub208 rep.pdf · Bifurcation analysis of chemical reaction mechanisms.

Heteroclinic bifurcations near Hopf-zero bifurcation in ...jswlamb/papers/LTW04.pdf · Heteroclinic bifurcations near Hopf-zero bifurcation in reversible vector fields inR3 Jeroen

Heteroclinic bifurcations near Hopf-zero bifurcation in ...jswlamb/papers/LTW04.pdf · Heteroclinic bifurcations near Hopf-zero bifurcation in reversible vector fields inR3 Jeroen

Hopf bifurcation of viscous shock waves in compressible gas

Hopf bifurcation of viscous shock waves in compressible gas

Reaction-Diffusion Models and Bifurcation Theory …jxshix.people.wm.edu/2013-taiwan/lecture10-taiwan.pdfReaction-Di usion Models and Bifurcation Theory Lecture 9: Hopf bifurcation

Reaction-Diffusion Models and Bifurcation Theory …jxshix.people.wm.edu/2013-taiwan/lecture10-taiwan.pdfReaction-Di usion Models and Bifurcation Theory Lecture 9: Hopf bifurcation

The Hopf Bifurcation and Its Applications - …authors.library.caltech.edu/25053/1/Hopf.pdfIn applications to partial differential equations, the ... validity of the Hopf bifurcation

The Hopf Bifurcation and Its Applications - …authors.library.caltech.edu/25053/1/Hopf.pdfIn applications to partial differential equations, the ... validity of the Hopf bifurcation

User guide on Hopf bifurcation and time periodic …User guide on Hopf bifurcation and time periodic orbits with pde2path Hannes Uecker Institut fur Mathematik, Universit at Oldenburg,

User guide on Hopf bifurcation and time periodic …User guide on Hopf bifurcation and time periodic orbits with pde2path Hannes Uecker Institut fur Mathematik, Universit at Oldenburg,

The Hopf-saddle-node bifurcation for fixed points of 3D ...

The Hopf-saddle-node bifurcation for fixed points of 3D ...

DYNAMICS OF STABILITY AND HOPF BIFURCATION ANALYSIS …

DYNAMICS OF STABILITY AND HOPF BIFURCATION ANALYSIS …

Research Article Hopf Bifurcation and Stability Analysis ...downloads.hindawi.com/journals/aaa/2014/632564.pdf · Research Article Hopf Bifurcation and Stability Analysis of a Congestion

Research Article Hopf Bifurcation and Stability Analysis ...downloads.hindawi.com/journals/aaa/2014/632564.pdf · Research Article Hopf Bifurcation and Stability Analysis of a Congestion

Hopf bifurcation and time periodic orbits with pde2path ... · Hopf bifurcation and time periodic orbits with pde2path { a tutorial via reaction{di usion systems and distributed optimal

Hopf bifurcation and time periodic orbits with pde2path ... · Hopf bifurcation and time periodic orbits with pde2path { a tutorial via reaction{di usion systems and distributed optimal

THE HOPF BIFURCATION AND ITS APPLICATIONS 163 · the hopf bifurcation and its applications 163 ... the hopf bifurcation and its applications 165 ... ceata + ceat a (2.9) with ...

THE HOPF BIFURCATION AND ITS APPLICATIONS 163 · the hopf bifurcation and its applications 163 ... the hopf bifurcation and its applications 165 ... ceata + ceat a (2.9) with ...

Global Asymptotic Stability and Hopf Bifurcation for a ...

Global Asymptotic Stability and Hopf Bifurcation for a ...

Generic Hopf bifurcation from lines of equilibria without …dynamics.mi.fu-berlin.de/preprints/FieLieAle98-BinOscill.pdf · Generic Hopf bifurcation from lines of equilibria without

Generic Hopf bifurcation from lines of equilibria without …dynamics.mi.fu-berlin.de/preprints/FieLieAle98-BinOscill.pdf · Generic Hopf bifurcation from lines of equilibria without