Exponential Stability and Periodicity of Fuzzy Delayed ...Exponential Stability and Periodicity of...
10
Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 645262, 9 pages http://dx.doi.org/10.1155/2013/645262 Research Article Exponential Stability and Periodicity of Fuzzy Delayed Reaction-Diffusion Cellular Neural Networks with Impulsive Effect Guowei Yang, 1 Yonggui Kao, 2 and Changhong Wang 2 1 College of Information Engineering, Nanchang Hangkong University, Nanchang 330063, China 2 Space Control and Inertial Technology Research Center, Harbin Institute of Technology, Harbin, Heilongjiang 150001, China Correspondence should be addressed to Changhong Wang; [email protected] Received 3 September 2012; Accepted 4 January 2013 Academic Editor: Tingwen Huang Copyright © 2013 Guowei Yang et al. 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. is paper considers dynamical behaviors of a class of fuzzy impulsive reaction-diffusion delayed cellular neural networks (FIRDDCNNs) with time-varying periodic self-inhibitions, interconnection weights, and inputs. By using delay differential inequality, M-matrix theory, and analytic methods, some new sufficient conditions ensuring global exponential stability of the periodic FIRDDCNN model with Neumann boundary conditions are established, and the exponential convergence rate index is estimated. e differentiability of the time-varying delays is not needed. An example is presented to demonstrate the efficiency and effectiveness of the obtained results. 1. Introduction e fuzzy cellular neural networks (FCNNs) model, which combines fuzzy logic with the structure of traditional neural networks (CNNs) [1–3], has been proposed by Yang et al. [4, 5]. Unlike previous CNNs structures, the FCNNs model has fuzzy logic between its template and input and/or output besides the “sum of product” operation. Studies have shown that the FCNNs model is a very useful paradigm for image processing and pattern recognition [6–8]. ese applications heavily depend on not only the dynamical analysis of equi- librium points but also on that of the periodic oscillatory solutions. In fact, the human brain is naturally in periodic oscillatory [9], and the dynamical analysis of periodic oscil- latory solutions is very important in learning theory [10, 11], because learning usually requires repetition. Moreover, an equilibrium point can be viewed as a special periodic solution of neural networks with arbitrary period. Stability analysis problems for FCNNs with and without delays have recently been probed; see [12–22] and the references therein. Yuan et al. [13] have investigated stability of FCNNs by linear matrix inequality approach, and several criteria have been provided for checking the periodic solutions for FCNNs with time- varying delays. Huang [14] has probed exponential stability of fuzzy cellular neural networks with distributed delay, without considering reaction-diffusion effects. Strictly speaking, reaction-diffusion effects cannot be neglected in both biological and man-made neural net- works [19–32], especially when electrons are moving in noneven electromagnetic field. In [19], stability is considered for FCNNs with diffusion terms and time-varying delay. Wang and Lu [20] have probed global exponential stability of FCNNs with delays and reaction-diffusion terms. Song and Wang [21] have studied dynamical behaviors of fuzzy reaction-diffusion periodic cellular neural networks with variable coefficients and delays without considering pulsing effects. Wang et al. [22] have discussed exponential stabil- ity of impulsive stochastic fuzzy reaction-diffusion Cohen- Grossberg neural networks with mixed delays. Zhao and Mao [30] have investigated boundedness and stability of nonau- tonomous cellular neural networks with reaction-diffusion terms. Zhao and Wang [31] have considered existence of periodic oscillatory solution of reaction-diffusion neural networks with delays without fuzzy logic and impulsive effect.
Transcript of Exponential Stability and Periodicity of Fuzzy Delayed ...Exponential Stability and Periodicity of...
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 645262 9 pageshttpdxdoiorg1011552013645262
Research ArticleExponential Stability and Periodicity ofFuzzy Delayed Reaction-Diffusion CellularNeural Networks with Impulsive Effect
Guowei Yang1 Yonggui Kao2 and Changhong Wang2
1 College of Information Engineering Nanchang Hangkong University Nanchang 330063 China2 Space Control and Inertial Technology Research Center Harbin Institute of Technology HarbinHeilongjiang 150001 China
Correspondence should be addressed to Changhong Wang chwanghiteducn
Received 3 September 2012 Accepted 4 January 2013
Academic Editor Tingwen Huang
Copyright copy 2013 Guowei Yang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
This paper considers dynamical behaviors of a class of fuzzy impulsive reaction-diffusion delayed cellular neural networks(FIRDDCNNs) with time-varying periodic self-inhibitions interconnection weights and inputs By using delay differentialinequality M-matrix theory and analytic methods some new sufficient conditions ensuring global exponential stability of theperiodic FIRDDCNN model with Neumann boundary conditions are established and the exponential convergence rate index isestimatedThe differentiability of the time-varying delays is not needed An example is presented to demonstrate the efficiency andeffectiveness of the obtained results
1 Introduction
The fuzzy cellular neural networks (FCNNs) model whichcombines fuzzy logic with the structure of traditional neuralnetworks (CNNs) [1ndash3] has been proposed by Yang et al[4 5] Unlike previous CNNs structures the FCNNs modelhas fuzzy logic between its template and input andor outputbesides the ldquosum of productrdquo operation Studies have shownthat the FCNNs model is a very useful paradigm for imageprocessing and pattern recognition [6ndash8] These applicationsheavily depend on not only the dynamical analysis of equi-librium points but also on that of the periodic oscillatorysolutions In fact the human brain is naturally in periodicoscillatory [9] and the dynamical analysis of periodic oscil-latory solutions is very important in learning theory [10 11]because learning usually requires repetition Moreover anequilibriumpoint can be viewed as a special periodic solutionof neural networks with arbitrary period Stability analysisproblems for FCNNs with and without delays have recentlybeen probed see [12ndash22] and the references therein Yuan etal [13] have investigated stability of FCNNs by linear matrixinequality approach and several criteria have been provided
for checking the periodic solutions for FCNNs with time-varying delays Huang [14] has probed exponential stability offuzzy cellular neural networks with distributed delay withoutconsidering reaction-diffusion effects
Strictly speaking reaction-diffusion effects cannot beneglected in both biological and man-made neural net-works [19ndash32] especially when electrons are moving innoneven electromagnetic field In [19] stability is consideredfor FCNNs with diffusion terms and time-varying delayWang and Lu [20] have probed global exponential stabilityof FCNNs with delays and reaction-diffusion terms Songand Wang [21] have studied dynamical behaviors of fuzzyreaction-diffusion periodic cellular neural networks withvariable coefficients and delays without considering pulsingeffects Wang et al [22] have discussed exponential stabil-ity of impulsive stochastic fuzzy reaction-diffusion Cohen-Grossberg neural networks withmixed delays Zhao andMao[30] have investigated boundedness and stability of nonau-tonomous cellular neural networks with reaction-diffusionterms Zhao and Wang [31] have considered existence ofperiodic oscillatory solution of reaction-diffusion neuralnetworkswith delayswithout fuzzy logic and impulsive effect
2 Abstract and Applied Analysis
As we all know many practical systems in physics biol-ogy engineering and information science undergo abruptchanges at certain moments of time because of impulsiveinputs [33] Impulsive differential equations and impulsiveneural networks have been received much interest in recentyears see for example [34ndash42] and the references thereinYang and Xu [36] have investigated existence and exponentialstability of periodic solution for impulsive delay differentialequations and applications Li and Lu [38] have discussedglobal exponential stability and existence of periodic solutionof Hopfield-type neural networks with impulses withoutreaction-diffusion To the best of our knowledge few authorshave probed the existence and exponential stability of theperiodic solutions for the FIRDDCNN model with variablecoefficients and time-varying delays As a result of thesimultaneous presence of fuzziness pulsing effects reaction-diffusion phenomena periodicity variable coefficients anddelays the dynamical behaviors of this kind ofmodel becomemuch more complex and have not been properly addressedwhich still remain important and challenging
Motivated by the above discussion we will establish somesufficient conditions for the existence and exponential stabil-ity of periodic solutions of this kind of FIRDDCNN modelapplying delay differential inequality 119872-matrix theory andanalytic methods An example is employed to demonstratethe usefulness of the obtained results
Notations Throughout this paper R119899 and R119899times119898 denoterespectively the 119899-dimensional Euclidean space and the setof all 119899 times119898 real matricesThe superscript ldquoTrdquo denotes matrixtransposition and the notation 119883 ge 119884 (resp 119883 gt 119884)where 119883 and 119884 are symmetric matrices means that 119883 minus 119884
is positive semidefinite (resp positive definite) Ω = 119909 =
(1199091 119909
119898)T |119909119894| lt 120583 is a bounded compact set in space
R119898 with smooth boundary 120597Ω and measure mesΩ gt 0Neumann boundary condition 120597119906
119894120597119899 = 0 is the outer
normal to 120597Ω 1198712(Ω) is the space of real functions Ω whichare 1198712 for the Lebesgue measure It is a Banach space with
the norm 119906(119905 119909)2
= (sum119899
119894=1119906119894(119905 119909)
2
2)12 where 119906(119905 119909) =
(1199061(119905 119909) 119906
119899(119905 119909))
T 119906119894(119905 119909)
2= (int
Ω
|119906119894(119905 119909)|
2
119889119909)12
|119906(119905 119909)| = (|1199061(119905 119909)| |119906
119899(119905 119909)|)
T For function 119892(119909) withpositive period 120596 we denote 119892 = max
119905isin[0120596]119892(119905) 119892 =
min119905isin[0120596]
119892(119905) Sometimes the arguments of a function or amatrix will be omitted in the analysis when no confusion canarise
2 Preliminaries
Consider the impulsive fuzzy reaction-diffusion delayed cel-lular neural networks (FIRDDCNN) model
120597119906119894(119905 119909)
120597119905=
119898
sum
119897=1
120597
120597119909119897
(119863119894
120597119906119894(119905 119909)
120597119909119897
)
minus 119888119894(119905) 119906119894(119905 119909) +
119899
sum
119895=1
119886119894119895(119905) 119891119895(119906119895(119905 119909))
+
119899
sum
119895=1
119887119894119895(119905) 119907119895(119905) + 119869
119894(119905)
+
119899
⋀
119895=1
120572119894119895(119905) 119892119895(119906119895(119905 minus 120591119895(119905) 119909))
+
119899
⋁
119895=1
120573119894119895(119905) 119892119895(119906119895(119905 minus 120591119895(119905) 119909))
+
119899
⋀
119895=1
119879119894119895(119905) 119907119895(119905) +
119899
⋁
119895=1
119867119894119895(119905) 119907119895(119905)
119905 = 119905119896 119909 isin Ω
119906119894(119905+
119896 119909) minus 119906
119894(119905minus
119896 119909) = 119868
119894119896(119906119894(119905minus
119896 119909))
119905 = 119905119896 119896 isin Z
+ 119909 isin Ω
120597119906119894(119905 119909)
120597119899= 0 119905 ge 119905
0 119909 isin 120597Ω
119906119894(1199050+ 119904 119909) = 120595
119894(119904 119909) minus120591
119895le 119904 le 0 119909 isin Ω
(1)
where 119899 ge 2 is the number of neurons in the network and119906119894(119905 119909) corresponds to the state of the 119894th neuron at time 119905
and in space 119909 119863 = diag(1198631 1198632 119863
119899) is the diffusion-
matrix and119863119894ge 0Δ = sum
119898
119896=1(1205972
1205971199092
119896) is the Laplace operator
119891119895(119906119895(119905 119909)) denotes the activation function of the 119895th unit
and 119907119895(119905) the activation function of the 119895th unit 119869
119894(119905) is an
input at time 119905 119888119894(119905) gt 0 represents the rate with which the
119894th unit will reset its potential to the resting state in isolationwhen disconnected from the networks and external inputsat time 119905 119886
119894119895(119905) and 119887
119894119895(119905) are elements of feedback template
and feed forward template at time 119905 respectivelyMoreover inmodel (1)120572
119894119895(119905) 120573119894119895(119905) 119879119894119895(119905) and119867
119894119895(119905) are elements of fuzzy
feedbackMIN template fuzzy feedbackMAX template fuzzyfeed forward MIN template and fuzzy feed forward MAXtemplate at time 119905 respectively the symbols ldquo⋀rdquo and ldquo⋁rdquodenote the fuzzy AND and fuzzy OR operation respectivelytime-varying delay 120591
119895(119905) is the transmission delay along the
axon of the 119895th unit and satisfies 0 le 120591119895(119905) le 120591
119895(120591119895is
a constant) the initial condition 120601119894(119904 119909) is bounded and
continuous on [minus120591 0] times Ω where 120591 = max1le119895le119899
120591119895 The fixed
moments 119905119896satisfy 0 = 119905
0lt 1199051
lt 1199052 lim
119896rarr+infin119905119896
=
+infin 119896 isin N 119906119894(119905+
119896 119909) and 119906
119894(119905minus
119896 119909) denote the right-hand
and left-hand limits at 119905119896 respectively We always assume
119906119894(119905+
119896 119909) = 119906
119894(119905119896 119909) for all 119896 isin 119873 The initial value functions
120595(119904 119909) belong to PCΩ([minus120591 0] times Ω 119877
119899
) PCΩ(119869 times Ω 119871
2
(Ω)) =
120595 119869timesΩ rarr 1198712
(Ω) | for every 119905 isin 119869 120595(119905 119909) isin 1198712
(Ω) for anyfixed 119909 isin Ω 120595(119905 119909) is continuous for all but atmost countablepoints 119904 isin 119869 and at these points 120595(119904+ 119909) and 120595(119904
minus
119909) exist120595(119904+
119909) = 120595(119904minus
119909) where 120595(119904+
119909) and 120595(119904minus
119909) denotethe right-hand and left-hand limit of the function 120595(119904 119909)respectively Especially let PC
Ω= PC([minus120591 0] times Ω 119871
2
(Ω))For any 120595(119905 119909) = (120595
1(119905 119909) 120595
119899(119905 119909)) isin PC
Ω suppose that
|120595119894(119905 119909)|
120591= sup
minus120591lt119904le0|120595119894(119905 + 119904 119909)| exists as a finite number
Abstract and Applied Analysis 3
and introduce the norm 120595(119905)2= (sum119899
119894=1120595119894(119905)2
2)12 where
120595119894(119905)2= (intΩ
|120595119894(119905 119909)|
2
119889119909)12
Throughout the paper we make the following assump-tions
(H1) There exists a positive diagonal matrix 119865 =
diag(1198651 1198652 119865
119899) and 119866 = diag(119866
1 1198662 119866
119899)
such that
119865119895= sup119909 = 119910
1003816100381610038161003816100381610038161003816100381610038161003816
119891119895(119909) minus 119891
119895(119910)
119909 minus 119910
1003816100381610038161003816100381610038161003816100381610038161003816
119866119895= sup119909 = 119910
1003816100381610038161003816100381610038161003816100381610038161003816
119892119895(119909) minus 119892
119895(119910)
119909 minus 119910
1003816100381610038161003816100381610038161003816100381610038161003816
(2)
for all 119909 = 119910 119895 = 1 2 119899
(H2) 119888119894(119905) gt 0 119886
119894119895(119905) 119887119894119895(119905) 120572119894119895(119905) 120573119894119895(119905) 119879119894119895(119905)119867119894119895(119905) 119907119894(119905)
119868119894(119905) and 120591
119895(119905) ge 0 are periodic function with a
common positive period 120596 for all 119905 ge 1199050 119894 119895 = 1
2 119899
(H3) For120596 gt 0 119894 = 1 2 119899 there exists 119902 isin 119885+such that
119905119896+ 120596 = 119905
119896+119902 119868119894119896(119906119894) = 119868119894(119896+119902)
(119906119894) and 119868
119894119896(119906119894(119905119896 119909)) are
Lipschitz continuous in R119899
Definition 1 Themodel in (1) is said to be globally exponen-tially periodic if (i) there exists one 120596-periodic solution and(ii) all other solutions of the model converge exponentially toit as 119905 rarr +infin
Definition 2 (see [26]) Let C = ([119905 minus 120591 119905]R119899) where 120591 ge
0 and 119865(119905 119909 119910) isin C(R+ times R119899 times CR119899) Then the function119865(119905 119909 119910) = (119891
1(119905 119909 119910) 119891
2(119905 119909 119910) 119891
119899(119905 119909 119910))
T is calledan 119872-function if (i) for every 119905 isin R+ 119909 isin R119899 119910(1) isin Cthere holds 119865(119905 119909 119910(1)) le 119865(119905 119909 119910
(2)
) for 119910(1)
le 119910(2) where
119910(1)
= (119910(1)
1 119910
(1)
119899)T and 119910
(2)
= (119910(2)
1 119910
(2)
119899)T (ii) every
119894th element of 119865 satisfies 119891119894(119905 119909(1)
119910) le 119891119894(119905 119909(2)
119910) for any119910 isin C 119905 ge 119905
0 where arbitrary 119909
(1) and 119909(2)
(119909(1)
le 119909(2)
)
belong to R119899 and have the same 119894th component 119909(1)119894
= 119909(2)
119894
Here 119909(1) = (119909(1)
1 119909
(1)
119899)T 119909(2)
= (119909(2)
1 119909
(2)
119899)T
Definition 3 (see [26]) A real matrix119860 = (119886119894119895)119899times119899
is said to bea nonsingular 119872-matrix if 119886
119894119895le 0 (119894 = 119895 119894 119895 = 1 119899) and
all successive principal minors of 119860 are positive
Lemma 4 (see [13]) Let 119906 and 119906lowast be two states of the model
in (1) then we have
10038161003816100381610038161003816100381610038161003816100381610038161003816
119899
⋀
119895=1
120572119894119895(119905) 119891119895(119906119895) minus
119899
⋀
119895=1
120572119894119895(119905) 119891119895(119906lowast
119895)
10038161003816100381610038161003816100381610038161003816100381610038161003816
le
119899
sum
119895=1
10038161003816100381610038161003816120572119894119895(119905)
10038161003816100381610038161003816sdot10038161003816100381610038161003816119891119895(119906119895) minus 119891119895(119906lowast
119895)10038161003816100381610038161003816
10038161003816100381610038161003816100381610038161003816100381610038161003816
119899
⋁
119895=1
120573119894119895(119905) 119891119895(119906119895) minus
119899
⋁
119895=1
120573119894119895(119905) 119891119895(119906lowast
119895)
10038161003816100381610038161003816100381610038161003816100381610038161003816
le
119899
sum
119895=1
10038161003816100381610038161003816120573119894119895(119905)
10038161003816100381610038161003816sdot10038161003816100381610038161003816119891119895(119906119895) minus 119891119895(119906lowast
119895)10038161003816100381610038161003816
(3)
Lemma 5 (see [26]) Assume that 119865(119905 119909 119910) is an119872-functionand (i) 119909(119905) lt 119910(119905) 119905 isin [119905 minus 120591 119905
0] (ii) 119863
+
119910(119905) gt
119865(119905 119910(119905) 119910119904
(119905)) 119863+119909(119905) le 119865(119905 119909(119905) 119909119904
(119905)) 119905 ge 1199050 where
119909119904
(119905) = supminus120591le119904le0
119909(119905 + 119904) 119910119904(119905) = supminus120591le119904le0
119910(119905 + 119904) Then119909(119905) lt 119910(119905) 119905 ge 119905
0
3 Main Results and Proofs
We should first point out that under assumptions (H1)(H2) and (H3) the FIRDDCNN model (1) has at least one120596-periodic solution of [26] The proof of the existence ofthe 120596-periodic solution of (1) can be carried out similar to[26 28] by the nonlinear functional analysis methods such astopological degree andhere is omittedWewillmainly discussthe uniqueness of the periodic solution and its exponentialstability
Theorem 6 Assume that (H1)ndash(H3) holds Furthermoreassume that the following conditions hold
(H4) C minus 119860F minus (120572 + 120573)119866 is a nonsingular119872-matrix
(H5) The impulsive operators ℎ119896(119906) = 119906 + 119868
119896(119906) is Lipschitz
continuous in R119899 that is there exists a nonnegativediagnose matrix Γ
119896= diag(1205741k 120574nk) such that
|ℎ119896(119906) minus ℎ
119896(119906lowast
)| le Γ119896|119906 minus 119906
lowast
| for all 119906 119906lowast isin R119899119896 isin 119873
+ where |ℎ119896(119906)| = (|ℎ
1119896(1199061)| |ℎ
119899119896(119906119899)|)
T119868119896(119906) = (119868
1119896(1199061) 119868
119899119896(119906119899))T
(H6) 120578 = sup119896isin119873+ln 120578119896(119905119896minus 119905119896minus1
) lt 120582 where 120578119896
=
max1le119894le119899
1 120574119894119896 119896 isin 119873
+
Then the model (1) is global exponential periodic andthe exponential convergence rate index 120582 minus 120578 and 120582 can beestimated by
120585119894(120582 minus 119888
119894)
+
119899
sum
119895=1
120585119895(10038161003816100381610038161003816119886119894119895
10038161003816100381610038161003816119865119895+ 119890120591120582
(10038161003816100381610038161003816120572119894119895
10038161003816100381610038161003816+10038161003816100381610038161003816120573119894119895
10038161003816100381610038161003816) 119866119895) lt 0
119894 = 1 119899
(4)
where 119862 = diag(1198881 119888
119899) and 120585
119894gt 0 119860 = (|119886
119894119895|)119899times119899
120572 =
(|120572119894119895|)119899times119899
120573 = (|120573119894119895|)119899times119899
satisfies minus120585119894119888119894+sum119899
119895=1120585119894(|119886119894119895|119865119894+
(|120572119894119895| + |120573119894119895|)119866119894) lt 0
Proof For any 120601 120595 isin PCΩ let 119906(119905 119909 120601) = (119906
1(119905 119909 120601)
119906119899(119905 119909 120601))
T be a periodic solution of the system (1) starting
4 Abstract and Applied Analysis
from 120601 and 119906(119905 119909 120595) = (1199061(119905 119909 120595) 119906
119899(119905 119909 120595))
T a solu-tion of the system (1) starting from 120595 Define
119906119905(120601 119909) = 119906 (119905 + 119904 119909 120601)
119906119905(120595 119909) = 119906 (119905 + 119904 119909 120595) 119904 isin [minus120591 0]
(5)
and we can see that 119906119905(120601 119909) 119906
119905(120595 119909) isin PC
Ωfor all 119905 gt 0 Let
119880119894= 119906119894(119905 119909 120601) minus 119906
119894(119905 119909 120595) then from (1) we get
120597119880119894
120597119905=
119898
sum
119897=1
120597
120597119909119897
(119863119894
120597119880119894
120597119909119897
) minus 119888119894(119905) 119880119894+
119899
sum
119895=1
119886119894119895(119905)
times [119891119895(119906119895(119905 119909 120601)) minus 119891
119895(119906119895(119905 119909 120595))]
+ [
[
119899
⋀
119895=1
120572119894119895(119905) 119891119895(119906119895(119905 minus 120591119895(119905) 119909 120601))
minus
119899
⋀
119895=1
120572119894119895(119905) 119891119895(119906119895(119905 minus 120591119895(119905) 119909 120595))]
]
+ [
[
119899
⋁
119895=1
120573119894119895(119905) 119891119895(119906119895(119905 minus 120591119895(119905) 119909 120601))
minus
119899
⋁
119895=1
120573119894119895(119905) 119891119895(119906119895(119905 minus 120591119895(119905) 119909 120595))]
]
(6)
for all 119905 = 119905119896 119909 isin Ω 119894 = 1 119899
Multiplying both sides of (6) by 119880119894and integrating it in
Ω we have1
2
dd119905
intΩ
1198802
119894d119909
= intΩ
119880119894
119898
sum
119897=1
120597
120597119909119897
(119863119894
120597119880119894
120597119909119897
) d119909
minus 119888119894(119905) intΩ
1198802
119894119889119909 +
119899
sum
119895=1
119886119894119895(119905) intΩ
119880119894
times [119891119895(119906119895(119905 119909 120601) minus 119891
119895(119906119895(119905 119909 120595)))] d119909
+ intΩ
119880119894
[
[
119899
⋀
119895=1
120572119894119895(119905) 119891119895(119906119895(119905 minus 120591119895(119905) 119909 120601))
minus
119899
⋀
119895=1
120572119894119895(119905) 119891119895(119906119895(119905 minus 120591119895(119905) 119909 120595))]
]
d119909
+ intΩ
119880119894
[
[
119899
⋁
119895=1
120573119894119895(119905) 119891119895(119906119895(119905 minus 120591119895(119905) 119909 120601))
minus
119899
⋁
119895=1
120573119894119895(119905) 119891119895(119906119895(119905 minus 120591119895(119905) 119909 120595))]
]
d119909
(7)
for 119905 = 119905119896 119909 isin Ω 119894 = 1 119899 By boundary condition and
Green Formula we can get
intΩ
119880119894
119898
sum
119897=1
120597
120597119909119897
(119863119894
120597119880119894
120597119909119897
)119889119909 le minus119863119894intΩ
(nabla119880119894)2d119909 (8)
Then from (8) (9) (H1)-(H2) Lemma 4 and the Holderinequality
119889
119889119905
10038171003817100381710038171198801198941003817100381710038171003817
2
2
le minus2119888119894
10038171003817100381710038171198801198941003817100381710038171003817
2
2+ 2
119899
sum
119895=1
10038161003816100381610038161003816119886119894119895
10038161003816100381610038161003816119865119895
100381710038171003817100381711988011989410038171003817100381710038172
10038171003817100381710038171003817119880119895
100381710038171003817100381710038172
+ 2
119899
sum
119895=1
(10038161003816100381610038161003816120572119894119895
10038161003816100381610038161003816+10038161003816100381610038161003816120573119894119895
10038161003816100381610038161003816) 119866119895
100381710038171003817100381711988011989410038171003817100381710038172
times10038171003817100381710038171003817119906119895(119905 minus 120591119895(119905) 119909 120601) minus 119906
119895(119905 minus 120591119895(119905) 119909 120595)
100381710038171003817100381710038172
119905 = 119905119896
(9)
Thus
119863+1003817100381710038171003817119880119894
10038171003817100381710038172
le minus119888119894
100381710038171003817100381711988011989410038171003817100381710038172
+
119899
sum
119895=1
10038161003816100381610038161003816120572119894119895
10038161003816100381610038161003816119865119895
10038171003817100381710038171003817119880119895
100381710038171003817100381710038172
+
119899
sum
119895=1
(10038161003816100381610038161003816120572119894119895
10038161003816100381610038161003816+10038161003816100381610038161003816120573119894119895
10038161003816100381610038161003816) 119866119895
times10038171003817100381710038171003817119906119895(119905 minus 120591119895(119905) 119909 120601) minus 119906
119895(119905 minus 120591119895(119905) 119909 120595)
100381710038171003817100381710038172
119905 = 119905119896
(10)
for 119894 = 1 119899 Since 119862 minus (119860119865 + (120572 + 120573)119866) is a nonsingular119872-matrix there exists a vector 120585 = (120585
1 120585
119899)Tgt 0 such that
minus120585119894119888119894+
119899
sum
119895=1
120585119895(10038161003816100381610038161003816119886119894119895
10038161003816100381610038161003816119865119895+ (
10038161003816100381610038161003816120572119894119895
10038161003816100381610038161003816+10038161003816100381610038161003816120573119894119895
10038161003816100381610038161003816) 119866119895) lt 0 (11)
Considering functions
Ψ119894(119910) = 120585
119894(119910 minus 119888
119894)
+
119899
sum
119895=1
120585119895(10038161003816100381610038161003816119886119894119895
10038161003816100381610038161003816119865119895+ 119890120591119910
(10038161003816100381610038161003816120572119894119895
10038161003816100381610038161003816+10038161003816100381610038161003816120573119894119895
10038161003816100381610038161003816) 119866119895)
119894 = 1 119899
(12)
we know from (11) that Ψ119894(0) lt 0 and Ψ
119894(119910) is continuous
Since dΨ119894(119910)d119910 gt 0 Ψ
119894(119910) is strictly monotonically
increasing there exists a scalar 120582119894gt 0 such that
Ψ119894(120582119894) = 120585119894(120582119894minus 119888119894)
+
119899
sum
119895=1
120585119895(10038161003816100381610038161003816119886119894119895
10038161003816100381610038161003816119865119895+ 119890120591120582119894 (10038161003816100381610038161003816120572119894119895
10038161003816100381610038161003816+10038161003816100381610038161003816120573119894119895
10038161003816100381610038161003816) 119866119895) = 0
119894 = 1 119899
(13)
Abstract and Applied Analysis 5
Choosing 0 lt 120582 lt min1205821 120582
119899 we have
120585119894(120582119894minus 119888119894)
+
119899
sum
119895=1
120585119895(10038161003816100381610038161003816119886119894119895
10038161003816100381610038161003816119865119895+ 119890120591120582119894 (10038161003816100381610038161003816120572119894119895
10038161003816100381610038161003816+10038161003816100381610038161003816120573119894119895
10038161003816100381610038161003816) 119866119895) lt 0
119894 = 1 119899
(14)
That is
120582120585 minus (119862 minus 119860119865) 120585 + (120572 + 120573)119866120585119890minus120582119905
lt 0 (15)
Furthermore choose a positive scalar 119901 large enough suchthat
119901119890minus120582119905
120585 gt (1 1 1)T 119905 isin [minus120591 0] (16)
For any 120576 gt 0 let
119903 (119905) = 119901119890minus120582119905
(1003817100381710038171003817120601 minus 120595
10038171003817100381710038172+ 120576) 120585 119905
0le 119905 lt 119905
1 (17)
From (15)ndash(17) we obtain
119863+
119903 (119905) gt minus (119862 minus 119860119865) 119903 (119905) + (120572 + 120573)119866119903119904
(119905)
= 119881 (119905 119903 (119905) 119903119904
(119905)) 1199050le 119905 lt 119905
1
(18)
where 119903119904(119905) = (119903119904
1(119905) 119903
119904
119899(119905))
T and 119903119904
119894(119905) = sup
minus120591le119904le0119901119890minus120582(119905+119904)
(120601 minus 1205932+ 120576)120585119894 It is easy to verify that 119881(119905 119903(119905) 119903
119904
(119905)) is an119872-function It follows also from (16) and (17) that
100381710038171003817100381711988011989410038171003817100381710038172
le1003817100381710038171003817120601 minus 120593
10038171003817100381710038172lt 119901119890minus120582119905
120585119894
1003817100381710038171003817120601 minus 12059310038171003817100381710038172
lt 119903119894(119905) 119905 isin [minus120591 0] 119894 = 1 2 119899
(19)
Denote
119880
= (10038171003817100381710038171199061 (119905 119909 120601) minus 119906
1(119905 119909 120595)
10038171003817100381710038172
1003817100381710038171003817119906119899 (119905 119909 120601) minus 119906119899(119905 119909 120595)
10038171003817100381710038172)T
119880(119904)
= (10038171003817100381710038171199061 (119905 119909 120601) minus 119906
1(119905 119909 120595)
1003817100381710038171003817
(119904)
2
1003817100381710038171003817119906119899 (119905 119909 120601) minus 119906119899(119905 119909 120595)
1003817100381710038171003817
(119904)
2)
T
(20)
where 119880119894(119904)
2= sup
minus120591le119904le0119906119894(119905 + 119904 119909 120601) minus 119906
119894(119905 + 119904 119909 120595)
2
then
119880
lt 119903 (119905) 119905 isin [minus120591 0] (21)
From (10) we can obtain
119863+
119880
le minus (119862 minus 119860119865)119880
+ (120572 + 120573)119866119880(119904)
= 119881 (119905 119880
119880(119904)
) 119905 = 119905119896
(22)
Now it follows from (18)ndash(22) and Lemma 5 that
119880
lt 119903 (119905) = 119901119890minus120582119905
(1003817100381710038171003817120601 minus 120595
10038171003817100381710038172+ 120576) 120585
1199050le 119905 lt 119905
1
(23)
Letting 120576 rarr 0 we have
119880
le 1199011205851003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890minus120582119905
1199050le 119905 lt 119905
1 (24)
And moreover from (24) we get
(
119899
sum
119894=1
10038171003817100381710038171198801198941003817100381710038171003817
2
2)
12
le 119901(
119899
sum
119894=1
1205852
119894)
12
1003817100381710038171003817120601 minus 12059510038171003817100381710038172119890minus120582119905
1199050le 119905 lt 119905
1
(25)
Let = 119901(sum119899
119894=11205852
119894)12 then ge 1 Define119882(119905) = 119906
119905(119909 120601)minus
119906119905(119909 120595)
2 it follows from (25) and the definitions of 119906
119905(120601 119909)
and 119906119905(120595 119909) that
119882(119905) =1003817100381710038171003817119906119905(119909 120601) minus 119906
119905(119909 120595)
10038171003817100381710038172
le 1003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890minus120582119905
1199050le 119905 lt 119905
1
(26)
It is easily observed that
119882(119905) le 1003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890minus120582119905
minus120591 le 119905 le 1199050= 0 (27)
Because (26) holds we can suppose that for 119897 le 119896 inequality
119882(119905) le 1205780sdot sdot sdot 120578119897minus1
1003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890minus120582119905
119905119897minus1
le 119905 lt 119905119897
(28)
holds where 1205780= 1 When 119897 = 119896 + 1 we note (H5) that
119882(119905119896) =
10038171003817100381710038171003817119906119905119896
(119909 120601) minus 119906119905119896
(119909 120595)100381710038171003817100381710038172
=10038171003817100381710038171003817ℎ119896(119906minus
119905119896
(119909 120601)) minus ℎ119896(119906minus
119905119896
(119909 120595))100381710038171003817100381710038172
le 120588 (Γ2
119896)10038171003817100381710038171003817119906minus
119905119896
(119909 120601) minus 119906minus
119905119896
(119909 120595)100381710038171003817100381710038172
= 120588 (Γ2
119896)119882 (119905
minus
119896)
le 1205780sdot sdot sdot 120578119897minus1
120588 (Γ2
119896)
1003817100381710038171003817120601 minus 12059510038171003817100381710038172119890minus120582119905119896
le 1205780sdot sdot sdot 120578119897minus1
120578119896120588 (Γ2
119896)
1003817100381710038171003817120601 minus 12059510038171003817100381710038172119890minus120582119905119896
(29)
where 120588(Γ2
119896) is the spectral radius of Γ
2
119896 Let 119872 =
max 120588(Γ2
119896) by (28) (29) and 120578 ge 1 we obtain
119882(119905) le 1205780sdot sdot sdot 120578119897minus1
120578119896119872
1003817100381710038171003817120601 minus 12059510038171003817100381710038172119890minus120582119905
119905119896minus 120591 le 119905 le 119905
119896
(30)
Combining (10) (17) (30) and Lemma 5 we get
119882(119905) le 1205780sdot sdot sdot 120578119897minus1
120578119896119872
1003817100381710038171003817120601 minus 12059510038171003817100381710038172119890minus120582119905
119905119896le 119905 lt 119905
119896+1 119896 isin 119873
+
(31)
Applying mathematical induction we conclude that
119882(119905) le 1205780sdot sdot sdot 120578119897minus1
1198721003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890minus120582119905
119905119896minus1
le 119905 lt 119905119896 119896 isin 119873
+
(32)
6 Abstract and Applied Analysis
From (H6) and (32) we have
119882(119905) le 1198901205781199051119890120578(1199052minus1199051)
sdot sdot sdot 119890120578(119905119896minus1minus119905119896minus2)
times1198721003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890minus120582119905
le 1198721003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890120578119905
119890minus120582119905
= 1198721003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890minus(120582minus120578)119905
119905119896minus1
le 119905 lt 119905119896 119896 isin 119873
+
(33)
This means that1003817100381710038171003817119906119905(119909 120601) minus 119906
119905(119909 120595)
10038171003817100381710038172
le 1198721003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890minus(120582minus120578)119905
le 1198721003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890minus(120582minus120578)(119905minus120591)
119905 ge 1199050
(34)
choosing a positive integer119873 such that
119872119890minus(120582minus120578)(119873120596minus120591)
le1
6 (35)
Define a Poincare mappingD Γ rarr Γ by
D (120601) = 119906120596(119909 120601) (36)
Then
D119873
(120601) = 119906119873120596
(119909 120601) (37)
Setting 119905 = 119873120596 in (34) from (35) and (37) we have
10038171003817100381710038171003817D119873
(120601) minusD119873
(120595)100381710038171003817100381710038172
le1
6
1003817100381710038171003817120601 minus 12059510038171003817100381710038172 (38)
which implies thatD119873 is a contraction mapping Thus thereexists a unique fixed point 120601lowast isin Γ such that
D119873
(D (120601lowast
)) = D (D119873
(120601lowast
)) = D (120601lowast
) (39)
From (37) we know that D(120601lowast
) is also a fixed point of D119873and then it follows from the uniqueness of the fixed point that
D (120601lowast
) = 120601lowast
that is 119906120596(119909 120601lowast
) = 120601lowast
(40)
Let 119906(119905 119909 120601lowast
) be a solution of the model (1) then 119906(119905 +
120596 119909 120601lowast
) is also a solution of the model (1) Obviously
119906119905+120596
(119909 120601lowast
) = 119906119905(119906120596(119909 120601lowast
)) = 119906119905(119909 120601lowast
) (41)
for all 119905 ge 1199050 Hence 119906(119905 + 120596 119909 120601
lowast
) = 119906(119905 119909 120601lowast
) whichshows that 119906(119905 119909 120601lowast) is exactly one 120596-periodic solution ofmodel (1) It is easy to see that all other solutions of model (1)converge to this periodic solution exponentially as 119905 rarr +infinand the exponential convergence rate index is 120582minus120578The proofis completed
Remark 7 When 119888119894(119905) = 119888
119894 119886119894119895(119905) = 119886
119894119895 119887119894119895(119905) = 119887
119894119895 120572119894119895(119905) = 120572
119894119895
120573119894119895(119905) = 120573
119894119895 119879119894119895(119905) = 119879
119894119895119867119894119895(119905) = 119867
119894119895 119907119894(119905) = 119907
119894 119868119894(119905) = 119868
119894 and
120591119905= 120591119894(119888119894 119886119894119895 119887119894119895 120572119894119895 120573119894119895 119879119894119895 119867119894119895 119907119894 119868119894 and 120591
119894are constants)
then the model (1) is changed into
120597119906119894(119905 119909)
120597119905=
119898
sum
119897=1
120597
120597119909119897
(119863119894
120597119906119894(119905 119909)
120597119909119897
)
minus 119888119894119906119894(119905 119909) +
119899
sum
119895=1
119886119894119895119891119895(119906119895(119905 119909))
+
119899
sum
119895=1
119887119894119895119907119895+ 119869119894
+
119899
⋀
119895=1
120572119894119895119892119895(119906119895(119905 minus 120591119895(119905) 119909))
+
119899
⋁
119895=1
120573119894119895119892119895(119906119895(119905 minus 120591119895(119905) 119909))
+
119899
⋀
119895=1
119879119894119895119907119895+
119899
⋁
119895=1
119867119894119895119907119895 119905 = 119905
119896 119909 isin Ω
119906119894(119905+
119896 119909) minus 119906
119894(119905minus
119896 119909) = 119868
119894119896(119906119894(119905minus
119896 119909))
119905 = 119905119896 119896 isin Z
+ 119909 isin Ω
120597119906119894(119905 119909)
120597119899= 0 119905 ge 119905
0 119909 isin 120597Ω
119906119894(1199050+ 119904 119909) = 120595
119894(119904 119909) minus120591
119895le 119904 le 0 119909 isin Ω
(42)
For any positive constant 120596 ge 0 we have 119888119894(119905 + 120596) = 119888
119894(119905)
119886119894119895(119905 + 120596) = 119886
119894119895(119905) 119887119894119895(119905 + 120596) = 119887
119894119895(119905) 120572119894119895(119905 + 120596) = 120572
119894119895(119905)
120573119894119895(119905 + 120596) = 120573
119894119895(119905) 119879119894119895(119905 + 120596) = 119879
119894119895(119905) 119867
119894119895(119905 + 120596) = 119867
119894119895(119905)
119907119894(119905+120596) = 119907
119894(119905) 119868119894(119905+120596) = 119868
119894(119905) and 120591
119894(119905+120596) = 120591
119894(119905) for 119905 ge 119905
0
Thus the sufficient conditions inTheorem 6 are satisfied
Remark 8 If 119868119896(sdot) = 0 the model (1) is changed into
120597119906119894(119905 119909)
120597119905=
119898
sum
119897=1
120597
120597119909119897
(119863119894
120597119906119894(119905 119909)
120597119909119897
)
minus 119888119894(119905) 119906119894(119905 119909) +
119899
sum
119895=1
119886119894119895(119905) 119891119895(119906119895(119905 119909))
+
119899
sum
119895=1
119887119894119895(119905) 119907119895(119905) + 119869
119894(119905)
+
119899
⋀
119895=1
120572119894119895(119905) 119892119895(119906119895(119905 minus 120591119895(119905) 119909))
+
119899
⋁
119895=1
120573119894119895(119905) 119892119895(119906119895(119905 minus 120591119895(119905) 119909))
Abstract and Applied Analysis 7
6
4
2
0minus5
05
1015
20 76
54
32
10
1199061
119909
119905
5 43
21
Figure 1 State response 1199061(119905 119909) of model (44) without impulsiveeffects
+
119899
⋀
119895=1
119879119894119895(119905) 119907119895(119905) +
119899
⋁
119895=1
119867119894119895(119905) 119907119895(119905)
119905 = 119905119896 119909 isin Ω
120597119906119894(119905 119909)
120597119899= 0 119905 ge 119905
0 119909 isin 120597Ω
119906119894(1199050+ 119904 119909) = 120595
119894(119904 119909) minus120591
119895le 119904 le 0 119909 isin Ω
(43)
which has been discussed in [22] As Song and Wang havepointed out the model (43) is more general than some well-studied fuzzy neural networks For example when 119888
119894(119905) gt
0 119886119894119895(119905) 119887119894119895(119905) 120572119894119895(119905) 120573119894119895(119905) 119879119894119895(119905)119867119894119895(119905) 119907119894(119905) and 119868
119894(119905) are all
constants the model in (43) reduces the model which hasbeen studied by Huang [19] Moreover if 119863
119894= 0 120591
119894(119905) = 0
119891119894(120579) = 119892
119894(120579) = (12)(|120579 + 1| minus |120579 minus 1|) (119894 = 1 119899) then
model (42) covers the model studied by Yang et al [4 5] as aspecial case If119863
119894= 0 and 120591
119895(119905) is assumed to be differentiable
for 119894 119895 = 1 2 119899 then model (43) can be specialized tothe model investigated in Liu and Tang [12] and Yuan et al[13] Obviously our results are less conservative than that ofthe above-mentioned literature because they do not considerimpulsive effects
4 Numerical Examples
Example 9 Consider a two-neuron FIRDDCNNmodel
120597119906119894(119905 119909)
120597119905=
119898
sum
119897=1
120597
120597119909119897
(119863119894
120597119906119894(119905 119909)
120597119909119897
)
minus 119888119894(119905) 119906119894(119905 119909)
6
4
2
0minus5
05
1015
20 76
54
32
10
1199061
119909
119905
Figure 2 State response 1199061(119905 119909) of model (44) with impulsiveeffects
+
2
sum
119895=1
119886119894119895(119905) 119891119895(119906119895(119905 119909))
+
2
sum
119895=1
119887119894119895(119905) 119907119895(119905) + 119869
119894(119905)
+
2
⋀
119895=1
120572119894119895(119905) 119892119895(119906119895(119905 minus 120591119895(119905) 119909))
+
2
⋁
119895=1
120573119894119895(119905) 119892119895(119906119895(119905 minus 120591119895(119905) 119909))
+
2
⋀
119895=1
119879119894119895(119905) 119907119895(119905) +
2
⋁
119895=1
119867119894119895(119905) 119907119895(119905)
119905 = 119905119896 119909 isin Ω
119906119894(119905+
119896 119909) = (1 minus 120574
119894119896) 119906119894(119905minus
119896 119909)
119905 = 119905119896 119896 isin Z
+ 119909 isin Ω
120597119906119894(119905 119909)
120597119899= 0 119905 ge 119905
0 119909 isin 120597Ω
119906119894(1199050+ 119904 119909) = 120595
119894(119904 119909)
minus 120591119895le 119904 le 0 119909 isin Ω
(44)
where 119894 = 1 2 1198881(119905) = 26 119888
2(119905) = 208 119886
11(119905) = minus1 minus cos(119905)
11988612(119905) = 1 + cos(119905) 119886
21(119905) = 1 + sin(119905) 119886
22(119905) = minus1 minus sin(119905)
1198631= 8119863
2= 4 120597119906
119894(119905 119909)120597119899 = 0 (119905 ge 119905
0 119909 = 0 2120587) 120574
1119896= 04
1205742119896
= 02 1205951(sdot) = 120595
1(sdot) = 5 119887
11(119905) = 119887
21(119905) = cos(119905) 119887
12(119905) =
11988722(119905) = minus cos(119905) 119869
1(119905) = 119869
2(119905) = 1 119867
11(119905) = 119867
21(119905) = sin(119905)
11986712(119905) = 119867
22(119905) = minus1 + sin(119905) 119879
11(119905) = 119879
21(119905) = minus sin(119905)
11987912(119905) = 119879
22(119905) = 2 + sin(119905) 120591
1(119905) = 120591
2(119905) = 1 119891
119895(119906119895) =
119906119895(119905 119909) (119895 = 1 2) 119892
119895(119906119895(119905 minus 1 119909)) = 119906
119895(119905 minus 1 119909)119890
minus119906119895(119905minus1119909)
(119895 =
1 2) 12057211(119905) = minus128 120572
21(119905) = 120572
12(119905) = minus1 + cos(119905) 120572
22(119905) =
minus10 12057311(119905) = 128 120573
12(119905) = minus1 + sin(119905) = 120573
21(119905) 12057322(119905) =
8 Abstract and Applied Analysis
6
4
2
0minus5
05
1015
20 76
54
32
10
1199062
119909
119905
5 43
21
119909
Figure 3 State response 1199062(119905 119909) of model (44) without impulsiveeffects
6
4
2
0minus5
05
1015
20 76
54
32
10
1199062
119909
119905
5 43
21
119909
Figure 4 State response 1199062(119905 119909) of model (44) with impulsiveeffects
10 119907119895(119905) = sin(119905) We assume that there exists 119902 = 6 such
that 119905119896+ 2120587 = 119905
119896+119902 Obviously 119891
1 1198912 1198921 and 119892
2satisfy the
assumption (H1) with 1198651= 1198652= 1198661= 1198662= 1 and (H2) and
(H3) are satisfied with a common positive period 2120587
119862 minus (119860 + 120572 + 120573)119865 =[[
[
2
50
04
5
]]
]
(45)
is a nonsingular 119872-matrix The conditions of Theorem 6 aresatisfied hence there exists exactly one 2120596-periodic solutionof the model and all other solutions of the model convergeexponentially to it as 119905 rarr +infin Furthermore the exponentialconverging index can be calculated as120582 = 0021 because here120578119896= 1 and 120578 = 0 The simulation results are shown in Figures
1 2 3 and 4 respectively
5 Conclusions
In this paper periodicity and global exponential stability ofa class of FIRDDCNN model with variable both coefficients
and delays have been investigated By using Halanayrsquos delaydifferential inequality 119872-matrix theory and analytic meth-ods some new sufficient conditions have been established toguarantee the existence uniqueness and global exponentialstability of the periodic solution Moreover the exponentialconvergence rate index can be estimated An example andits simulation have been given to show the effectiveness ofthe obtained results In particular the differentiability ofthe time-varying delays has been removed The dynamicbehaviors of fuzzy neural networks with the property ofexponential periodicity are of great importance inmany areassuch as learning systems
Acknowledgments
The authors would like to thank the editors and the anony-mous reviewers for their valuable comments and construc-tive suggestions This research is supported by the NaturalScience Foundation of Guangxi Autonomous Region (no2012GXNSFBA053003) the National Natural Science Foun-dations of China (60973048 61272077 60974025 60673101and 60939003) National 863 Plan Project (2008 AA04Z4012009 AA043404) the Natural Science Foundation of Shan-dong Province (no Y2007G30) the Scientific andTechnolog-ical Project of Shandong Province (no 2007GG3WZ04016)the Science Foundation of Harbin Institute of Technology(Weihai) (HIT (WH) 200807) theNatural ScientificResearchInnovation Foundation in Harbin Institute of Technology(HIT NSRIF 2001120) the China Postdoctoral ScienceFoundation (2010048 1000) and the Shandong ProvincialKey Laboratory of Industrial Control Technique (QingdaoUniversity)
References
[1] LOChua andL Yang ldquoCellular neural networks theoryrdquo IEEETransactions on Circuits and Systems vol 35 no 10 pp 1257ndash1272 1988
[2] L O Chua and L Yang ldquoCellular neural networks applica-tionsrdquo IEEE Transactions on Circuits and Systems vol 35 no10 pp 1273ndash1290 1988
[3] T Roska and L O Chua ldquoCellular neural networks withnon-linear and delay-type template elements and non-uniformgridsrdquo International Journal of Circuit Theory and Applicationsvol 20 no 5 pp 469ndash481 1992
[4] T Yang L B Yang C W Wu and L O Chua ldquoFuzzycellular neural networks theoryrdquo in Proceedings of the 4th IEEEInternational Workshop on Cellular Neural Networks and TheirApplications (CNNA rsquo96) pp 181ndash186 June 1996
[5] T Yang and L-B Yang ldquoThe global stability of fuzzy cellularneural networkrdquo IEEE Transactions on Circuits and Systems Ivol 43 no 10 pp 880ndash883 1996
[6] T Yang L B Yang C W Wu and L O Chua ldquoFuzzy cellularneural networks applicationsrdquo in Proceedings of the 4th IEEEInternational Workshop on Cellular Neural Networks and TheirApplications (CNNA rsquo96) pp 225ndash230 June 1996
[7] S T Wang K F L Chung and F Duan ldquoApplying theimproved fuzzy cellular neural network IFCNN to white bloodcell detectionrdquo Neurocomputing vol 70 no 7ndash9 pp 1348ndash13592007
Abstract and Applied Analysis 9
[8] T G Barbounis and J B Theocharis ldquoA locally recurrent fuzzyneural network with application to the wind speed predictionusing spatial correlationrdquo Neurocomputing vol 70 no 7ndash9 pp1525ndash1542 2007
[9] H Zhao ldquoGlobal exponential stability and periodicity of cellularneural networks with variable delaysrdquoPhysics Letters A vol 336no 4-5 pp 331ndash341 2005
[10] S Townley A Ilchmann M G Weiss et al ldquoExistence andlearning of oscillations in recurrent neural networksrdquo IEEETransactions on Neural Networks vol 11 no 1 pp 205ndash2142000
[11] Z Huang and Y Xia ldquoExponential periodic attractor ofimpulsive BAM networks with finite distributed delaysrdquo ChaosSolitons amp Fractals vol 39 no 1 pp 373ndash384 2009
[12] Y Liu and W Tang ldquoExponential stability of fuzzy cellularneural networkswith constant and time-varying delaysrdquoPhysicsLetters A vol 323 no 3-4 pp 224ndash233 2004
[13] K Yuan J Cao and J Deng ldquoExponential stability and periodicsolutions of fuzzy cellular neural networks with time-varyingdelaysrdquoNeurocomputing vol 69 no 13ndash15 pp 1619ndash1627 2006
[14] T Huang ldquoExponential stability of fuzzy cellular neural net-works with distributed delayrdquo Physics Letters A vol 351 no 1-2pp 48ndash52 2006
[15] G Bao S Wen and Z Zeng ldquoRobust stability analysis of inter-val fuzzy Cohen-Grossberg neural networks with piecewiseconstant argument of generalized typerdquo Neural Networks vol33 pp 32ndash41 2012
[16] H Zhao N Ding and L Chen ldquoAlmost sure exponential sta-bility of stochastic fuzzy cellular neural networks with delaysrdquoChaos Solitons amp Fractals vol 40 no 4 pp 1653ndash1659 2009
[17] W Ding M Han and M Li ldquoExponential lag synchronizationof delayed fuzzy cellular neural networkswith impulsesrdquoPhysicsLetters A vol 373 no 8-9 pp 832ndash837 2009
[18] L Chen and H Zhao ldquoStability analysis of stochastic fuzzycellular neural networks with delaysrdquo Neurocomputing vol 72no 1ndash3 pp 436ndash444 2008
[19] T Huang ldquoExponential stability of delayed fuzzy cellular neuralnetworks with diffusionrdquo Chaos Solitons amp Fractals vol 31 no3 pp 658ndash664 2007
[20] J Wang and J G Lu ldquoGlobal exponential stability of fuzzycellular neural networks with delays and reaction-diffusiontermsrdquo Chaos Solitons amp Fractals vol 38 no 3 pp 878ndash8852008
[21] Q Song and Z Wang ldquoDynamical behaviors of fuzzy reaction-diffusion periodic cellular neural networks with variable coeffi-cients and delaysrdquo Applied Mathematical Modelling vol 33 no9 pp 3533ndash3545 2009
[22] C Wang Y kao and G Yang ldquoExponential stability ofimpulsive stochastic fuzzy reaction-diffusion Cohen-Grossbergneural networks with mixed delaysrdquo Neurocomputing vol 89pp 55ndash63 2012
[23] T Serrano-Gotarredona and B Linares-Barranco ldquoLog-Domain Implementation of Complex Dynamics Reaction-Diffusion Neural Networksrdquo IEEE Transactions on NeuralNetworks vol 14 no 5 pp 1337ndash1355 2003
[24] J Liang and J Cao ldquoGlobal exponential stability of reaction-diffusion recurrent neural networks with time-varying delaysrdquoPhysics Letters A vol 314 no 5-6 pp 434ndash442 2003
[25] L Wang and D Xu ldquoAsymptotic behavior of a class of reaction-diffusion equations with delaysrdquo Journal of Mathematical Anal-ysis and Applications vol 281 no 2 pp 439ndash453 2003
[26] L Wang and D Xu ldquoGlobal exponential stability of Hopfieldreaction-diffusion neural networks with time-varying delaysrdquoScience in China Series F vol 46 no 6 pp 466ndash474 2003
[27] Z Yang and D Xu ldquoGlobal dynamics for non-autonomousreaction-diffusion neural networks with time-varying delaysrdquoTheoretical Computer Science vol 403 no 1 pp 3ndash10 2008
[28] Y G Kao C C Gao and D K Wang ldquoGlobal exponentialstability of reaction-diffusion Hopfield neural networks withcontinuously distributed delaysrdquo Mathematica Applicata vol21 no 3 pp 457ndash462 2008 (Chinese)
[29] Y-G Kao J-F Guo C-H Wang and X-Q Sun ldquoDelay-dependent robust exponential stability of Markovian jump-ing reaction-diffusion Cohen-Grossberg neural networks withmixed delaysrdquo Journal of the Franklin Institute vol 349 no 6pp 1972ndash1988 2012
[30] H Zhao and Z Mao ldquoBoundedness and stability of nonau-tonomous cellular neural networks with reaction-diffusiontermsrdquo Mathematics and Computers in Simulation vol 79 no5 pp 1603ndash1617 2009
[31] H Zhao and G Wang ldquoExistence of periodic oscillatorysolution of reaction-diffusion neural networks with delaysrdquoPhysics Letters A vol 343 no 5 pp 372ndash383 2005
[32] Q Gan ldquoExponential synchronization of stochastic Cohen-Grossberg neural networks withmixed time-varying delays andreaction-diffusion via periodically intermittent controlrdquoNeuralNetworks vol 31 pp 12ndash21 2012
[33] V Lakshmikantham D D Baınov and P S Simeonov Theoryof Impulsive Differential Equations vol 6 of Series in ModernApplied Mathematics World Scientific Teaneck NJ USA 1989
[34] D D Bainov and P S Simeonov Impulsive Differential Equa-tions Periodic Solutions and Applications Longman HarlowUK 1993
[35] X Song X Xin andWHuang ldquoExponential stability of delayedand impulsive cellular neural networks with partially Lipschitzcontinuous activation functionsrdquo Neural Networks vol 29-30pp 80ndash90 2012
[36] Z Yang and D Xu ldquoExistence and exponential stability ofperiodic solution for impulsive delay differential equations andapplicationsrdquo Nonlinear Analysis Theory Methods amp Applica-tions vol 64 no 1 pp 130ndash145 2006
[37] X Yang X Liao D J Evans and Y Tang ldquoExistence andstability of periodic solution in impulsive Hopfield neuralnetworks with finite distributed delaysrdquo Physics Letters A vol343 no 1ndash3 pp 108ndash116 2005
[38] Y Li and L Lu ldquoGlobal exponential stability and existenceof periodic solution of Hopfield-type neural networks withimpulsesrdquo Physics Letters A vol 333 no 1-2 pp 62ndash71 2004
[39] Y Li L Zhu andP Liu ldquoExistence and stability of periodic solu-tions of delayed cellular neural networksrdquo Nonlinear AnalysisReal World Applications vol 7 no 2 pp 225ndash234 2006
[40] Z H Guan and G Chen ldquoOn delayed impulsive Hopfieldneural networksrdquo Neural Networks vol 12 no 2 pp 273ndash2801999
[41] Z H Guan J Lam andG Chen ldquoOn impulsive autoassociativeneural networksrdquo Neural Networks vol 13 no 1 pp 63ndash692000
[42] W Ding ldquoSynchronization of delayed fuzzy cellular neuralnetworks with impulsive effectsrdquo Communications in NonlinearScience and Numerical Simulation vol 14 no 11 pp 3945ndash39522009
Submit your manuscripts athttpwwwhindawicom
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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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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
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Abstract and Applied Analysis
As we all know many practical systems in physics biol-ogy engineering and information science undergo abruptchanges at certain moments of time because of impulsiveinputs [33] Impulsive differential equations and impulsiveneural networks have been received much interest in recentyears see for example [34ndash42] and the references thereinYang and Xu [36] have investigated existence and exponentialstability of periodic solution for impulsive delay differentialequations and applications Li and Lu [38] have discussedglobal exponential stability and existence of periodic solutionof Hopfield-type neural networks with impulses withoutreaction-diffusion To the best of our knowledge few authorshave probed the existence and exponential stability of theperiodic solutions for the FIRDDCNN model with variablecoefficients and time-varying delays As a result of thesimultaneous presence of fuzziness pulsing effects reaction-diffusion phenomena periodicity variable coefficients anddelays the dynamical behaviors of this kind ofmodel becomemuch more complex and have not been properly addressedwhich still remain important and challenging
Motivated by the above discussion we will establish somesufficient conditions for the existence and exponential stabil-ity of periodic solutions of this kind of FIRDDCNN modelapplying delay differential inequality 119872-matrix theory andanalytic methods An example is employed to demonstratethe usefulness of the obtained results
Notations Throughout this paper R119899 and R119899times119898 denoterespectively the 119899-dimensional Euclidean space and the setof all 119899 times119898 real matricesThe superscript ldquoTrdquo denotes matrixtransposition and the notation 119883 ge 119884 (resp 119883 gt 119884)where 119883 and 119884 are symmetric matrices means that 119883 minus 119884
is positive semidefinite (resp positive definite) Ω = 119909 =
(1199091 119909
119898)T |119909119894| lt 120583 is a bounded compact set in space
R119898 with smooth boundary 120597Ω and measure mesΩ gt 0Neumann boundary condition 120597119906
119894120597119899 = 0 is the outer
normal to 120597Ω 1198712(Ω) is the space of real functions Ω whichare 1198712 for the Lebesgue measure It is a Banach space with
the norm 119906(119905 119909)2
= (sum119899
119894=1119906119894(119905 119909)
2
2)12 where 119906(119905 119909) =
(1199061(119905 119909) 119906
119899(119905 119909))
T 119906119894(119905 119909)
2= (int
Ω
|119906119894(119905 119909)|
2
119889119909)12
|119906(119905 119909)| = (|1199061(119905 119909)| |119906
119899(119905 119909)|)
T For function 119892(119909) withpositive period 120596 we denote 119892 = max
119905isin[0120596]119892(119905) 119892 =
min119905isin[0120596]
119892(119905) Sometimes the arguments of a function or amatrix will be omitted in the analysis when no confusion canarise
2 Preliminaries
Consider the impulsive fuzzy reaction-diffusion delayed cel-lular neural networks (FIRDDCNN) model
120597119906119894(119905 119909)
120597119905=
119898
sum
119897=1
120597
120597119909119897
(119863119894
120597119906119894(119905 119909)
120597119909119897
)
minus 119888119894(119905) 119906119894(119905 119909) +
119899
sum
119895=1
119886119894119895(119905) 119891119895(119906119895(119905 119909))
+
119899
sum
119895=1
119887119894119895(119905) 119907119895(119905) + 119869
119894(119905)
+
119899
⋀
119895=1
120572119894119895(119905) 119892119895(119906119895(119905 minus 120591119895(119905) 119909))
+
119899
⋁
119895=1
120573119894119895(119905) 119892119895(119906119895(119905 minus 120591119895(119905) 119909))
+
119899
⋀
119895=1
119879119894119895(119905) 119907119895(119905) +
119899
⋁
119895=1
119867119894119895(119905) 119907119895(119905)
119905 = 119905119896 119909 isin Ω
119906119894(119905+
119896 119909) minus 119906
119894(119905minus
119896 119909) = 119868
119894119896(119906119894(119905minus
119896 119909))
119905 = 119905119896 119896 isin Z
+ 119909 isin Ω
120597119906119894(119905 119909)
120597119899= 0 119905 ge 119905
0 119909 isin 120597Ω
119906119894(1199050+ 119904 119909) = 120595
119894(119904 119909) minus120591
119895le 119904 le 0 119909 isin Ω
(1)
where 119899 ge 2 is the number of neurons in the network and119906119894(119905 119909) corresponds to the state of the 119894th neuron at time 119905
and in space 119909 119863 = diag(1198631 1198632 119863
119899) is the diffusion-
matrix and119863119894ge 0Δ = sum
119898
119896=1(1205972
1205971199092
119896) is the Laplace operator
119891119895(119906119895(119905 119909)) denotes the activation function of the 119895th unit
and 119907119895(119905) the activation function of the 119895th unit 119869
119894(119905) is an
input at time 119905 119888119894(119905) gt 0 represents the rate with which the
119894th unit will reset its potential to the resting state in isolationwhen disconnected from the networks and external inputsat time 119905 119886
119894119895(119905) and 119887
119894119895(119905) are elements of feedback template
and feed forward template at time 119905 respectivelyMoreover inmodel (1)120572
119894119895(119905) 120573119894119895(119905) 119879119894119895(119905) and119867
119894119895(119905) are elements of fuzzy
feedbackMIN template fuzzy feedbackMAX template fuzzyfeed forward MIN template and fuzzy feed forward MAXtemplate at time 119905 respectively the symbols ldquo⋀rdquo and ldquo⋁rdquodenote the fuzzy AND and fuzzy OR operation respectivelytime-varying delay 120591
119895(119905) is the transmission delay along the
axon of the 119895th unit and satisfies 0 le 120591119895(119905) le 120591
119895(120591119895is
a constant) the initial condition 120601119894(119904 119909) is bounded and
continuous on [minus120591 0] times Ω where 120591 = max1le119895le119899
120591119895 The fixed
moments 119905119896satisfy 0 = 119905
0lt 1199051
lt 1199052 lim
119896rarr+infin119905119896
=
+infin 119896 isin N 119906119894(119905+
119896 119909) and 119906
119894(119905minus
119896 119909) denote the right-hand
and left-hand limits at 119905119896 respectively We always assume
119906119894(119905+
119896 119909) = 119906
119894(119905119896 119909) for all 119896 isin 119873 The initial value functions
120595(119904 119909) belong to PCΩ([minus120591 0] times Ω 119877
119899
) PCΩ(119869 times Ω 119871
2
(Ω)) =
120595 119869timesΩ rarr 1198712
(Ω) | for every 119905 isin 119869 120595(119905 119909) isin 1198712
(Ω) for anyfixed 119909 isin Ω 120595(119905 119909) is continuous for all but atmost countablepoints 119904 isin 119869 and at these points 120595(119904+ 119909) and 120595(119904
minus
119909) exist120595(119904+
119909) = 120595(119904minus
119909) where 120595(119904+
119909) and 120595(119904minus
119909) denotethe right-hand and left-hand limit of the function 120595(119904 119909)respectively Especially let PC
Ω= PC([minus120591 0] times Ω 119871
2
(Ω))For any 120595(119905 119909) = (120595
1(119905 119909) 120595
119899(119905 119909)) isin PC
Ω suppose that
|120595119894(119905 119909)|
120591= sup
minus120591lt119904le0|120595119894(119905 + 119904 119909)| exists as a finite number
Abstract and Applied Analysis 3
and introduce the norm 120595(119905)2= (sum119899
119894=1120595119894(119905)2
2)12 where
120595119894(119905)2= (intΩ
|120595119894(119905 119909)|
2
119889119909)12
Throughout the paper we make the following assump-tions
(H1) There exists a positive diagonal matrix 119865 =
diag(1198651 1198652 119865
119899) and 119866 = diag(119866
1 1198662 119866
119899)
such that
119865119895= sup119909 = 119910
1003816100381610038161003816100381610038161003816100381610038161003816
119891119895(119909) minus 119891
119895(119910)
119909 minus 119910
1003816100381610038161003816100381610038161003816100381610038161003816
119866119895= sup119909 = 119910
1003816100381610038161003816100381610038161003816100381610038161003816
119892119895(119909) minus 119892
119895(119910)
119909 minus 119910
1003816100381610038161003816100381610038161003816100381610038161003816
(2)
for all 119909 = 119910 119895 = 1 2 119899
(H2) 119888119894(119905) gt 0 119886
119894119895(119905) 119887119894119895(119905) 120572119894119895(119905) 120573119894119895(119905) 119879119894119895(119905)119867119894119895(119905) 119907119894(119905)
119868119894(119905) and 120591
119895(119905) ge 0 are periodic function with a
common positive period 120596 for all 119905 ge 1199050 119894 119895 = 1
2 119899
(H3) For120596 gt 0 119894 = 1 2 119899 there exists 119902 isin 119885+such that
119905119896+ 120596 = 119905
119896+119902 119868119894119896(119906119894) = 119868119894(119896+119902)
(119906119894) and 119868
119894119896(119906119894(119905119896 119909)) are
Lipschitz continuous in R119899
Definition 1 Themodel in (1) is said to be globally exponen-tially periodic if (i) there exists one 120596-periodic solution and(ii) all other solutions of the model converge exponentially toit as 119905 rarr +infin
Definition 2 (see [26]) Let C = ([119905 minus 120591 119905]R119899) where 120591 ge
0 and 119865(119905 119909 119910) isin C(R+ times R119899 times CR119899) Then the function119865(119905 119909 119910) = (119891
1(119905 119909 119910) 119891
2(119905 119909 119910) 119891
119899(119905 119909 119910))
T is calledan 119872-function if (i) for every 119905 isin R+ 119909 isin R119899 119910(1) isin Cthere holds 119865(119905 119909 119910(1)) le 119865(119905 119909 119910
(2)
) for 119910(1)
le 119910(2) where
119910(1)
= (119910(1)
1 119910
(1)
119899)T and 119910
(2)
= (119910(2)
1 119910
(2)
119899)T (ii) every
119894th element of 119865 satisfies 119891119894(119905 119909(1)
119910) le 119891119894(119905 119909(2)
119910) for any119910 isin C 119905 ge 119905
0 where arbitrary 119909
(1) and 119909(2)
(119909(1)
le 119909(2)
)
belong to R119899 and have the same 119894th component 119909(1)119894
= 119909(2)
119894
Here 119909(1) = (119909(1)
1 119909
(1)
119899)T 119909(2)
= (119909(2)
1 119909
(2)
119899)T
Definition 3 (see [26]) A real matrix119860 = (119886119894119895)119899times119899
is said to bea nonsingular 119872-matrix if 119886
119894119895le 0 (119894 = 119895 119894 119895 = 1 119899) and
all successive principal minors of 119860 are positive
Lemma 4 (see [13]) Let 119906 and 119906lowast be two states of the model
in (1) then we have
10038161003816100381610038161003816100381610038161003816100381610038161003816
119899
⋀
119895=1
120572119894119895(119905) 119891119895(119906119895) minus
119899
⋀
119895=1
120572119894119895(119905) 119891119895(119906lowast
119895)
10038161003816100381610038161003816100381610038161003816100381610038161003816
le
119899
sum
119895=1
10038161003816100381610038161003816120572119894119895(119905)
10038161003816100381610038161003816sdot10038161003816100381610038161003816119891119895(119906119895) minus 119891119895(119906lowast
119895)10038161003816100381610038161003816
10038161003816100381610038161003816100381610038161003816100381610038161003816
119899
⋁
119895=1
120573119894119895(119905) 119891119895(119906119895) minus
119899
⋁
119895=1
120573119894119895(119905) 119891119895(119906lowast
119895)
10038161003816100381610038161003816100381610038161003816100381610038161003816
le
119899
sum
119895=1
10038161003816100381610038161003816120573119894119895(119905)
10038161003816100381610038161003816sdot10038161003816100381610038161003816119891119895(119906119895) minus 119891119895(119906lowast
119895)10038161003816100381610038161003816
(3)
Lemma 5 (see [26]) Assume that 119865(119905 119909 119910) is an119872-functionand (i) 119909(119905) lt 119910(119905) 119905 isin [119905 minus 120591 119905
0] (ii) 119863
+
119910(119905) gt
119865(119905 119910(119905) 119910119904
(119905)) 119863+119909(119905) le 119865(119905 119909(119905) 119909119904
(119905)) 119905 ge 1199050 where
119909119904
(119905) = supminus120591le119904le0
119909(119905 + 119904) 119910119904(119905) = supminus120591le119904le0
119910(119905 + 119904) Then119909(119905) lt 119910(119905) 119905 ge 119905
0
3 Main Results and Proofs
We should first point out that under assumptions (H1)(H2) and (H3) the FIRDDCNN model (1) has at least one120596-periodic solution of [26] The proof of the existence ofthe 120596-periodic solution of (1) can be carried out similar to[26 28] by the nonlinear functional analysis methods such astopological degree andhere is omittedWewillmainly discussthe uniqueness of the periodic solution and its exponentialstability
Theorem 6 Assume that (H1)ndash(H3) holds Furthermoreassume that the following conditions hold
(H4) C minus 119860F minus (120572 + 120573)119866 is a nonsingular119872-matrix
(H5) The impulsive operators ℎ119896(119906) = 119906 + 119868
119896(119906) is Lipschitz
continuous in R119899 that is there exists a nonnegativediagnose matrix Γ
119896= diag(1205741k 120574nk) such that
|ℎ119896(119906) minus ℎ
119896(119906lowast
)| le Γ119896|119906 minus 119906
lowast
| for all 119906 119906lowast isin R119899119896 isin 119873
+ where |ℎ119896(119906)| = (|ℎ
1119896(1199061)| |ℎ
119899119896(119906119899)|)
T119868119896(119906) = (119868
1119896(1199061) 119868
119899119896(119906119899))T
(H6) 120578 = sup119896isin119873+ln 120578119896(119905119896minus 119905119896minus1
) lt 120582 where 120578119896
=
max1le119894le119899
1 120574119894119896 119896 isin 119873
+
Then the model (1) is global exponential periodic andthe exponential convergence rate index 120582 minus 120578 and 120582 can beestimated by
120585119894(120582 minus 119888
119894)
+
119899
sum
119895=1
120585119895(10038161003816100381610038161003816119886119894119895
10038161003816100381610038161003816119865119895+ 119890120591120582
(10038161003816100381610038161003816120572119894119895
10038161003816100381610038161003816+10038161003816100381610038161003816120573119894119895
10038161003816100381610038161003816) 119866119895) lt 0
119894 = 1 119899
(4)
where 119862 = diag(1198881 119888
119899) and 120585
119894gt 0 119860 = (|119886
119894119895|)119899times119899
120572 =
(|120572119894119895|)119899times119899
120573 = (|120573119894119895|)119899times119899
satisfies minus120585119894119888119894+sum119899
119895=1120585119894(|119886119894119895|119865119894+
(|120572119894119895| + |120573119894119895|)119866119894) lt 0
Proof For any 120601 120595 isin PCΩ let 119906(119905 119909 120601) = (119906
1(119905 119909 120601)
119906119899(119905 119909 120601))
T be a periodic solution of the system (1) starting
4 Abstract and Applied Analysis
from 120601 and 119906(119905 119909 120595) = (1199061(119905 119909 120595) 119906
119899(119905 119909 120595))
T a solu-tion of the system (1) starting from 120595 Define
119906119905(120601 119909) = 119906 (119905 + 119904 119909 120601)
119906119905(120595 119909) = 119906 (119905 + 119904 119909 120595) 119904 isin [minus120591 0]
(5)
and we can see that 119906119905(120601 119909) 119906
119905(120595 119909) isin PC
Ωfor all 119905 gt 0 Let
119880119894= 119906119894(119905 119909 120601) minus 119906
119894(119905 119909 120595) then from (1) we get
120597119880119894
120597119905=
119898
sum
119897=1
120597
120597119909119897
(119863119894
120597119880119894
120597119909119897
) minus 119888119894(119905) 119880119894+
119899
sum
119895=1
119886119894119895(119905)
times [119891119895(119906119895(119905 119909 120601)) minus 119891
119895(119906119895(119905 119909 120595))]
+ [
[
119899
⋀
119895=1
120572119894119895(119905) 119891119895(119906119895(119905 minus 120591119895(119905) 119909 120601))
minus
119899
⋀
119895=1
120572119894119895(119905) 119891119895(119906119895(119905 minus 120591119895(119905) 119909 120595))]
]
+ [
[
119899
⋁
119895=1
120573119894119895(119905) 119891119895(119906119895(119905 minus 120591119895(119905) 119909 120601))
minus
119899
⋁
119895=1
120573119894119895(119905) 119891119895(119906119895(119905 minus 120591119895(119905) 119909 120595))]
]
(6)
for all 119905 = 119905119896 119909 isin Ω 119894 = 1 119899
Multiplying both sides of (6) by 119880119894and integrating it in
Ω we have1
2
dd119905
intΩ
1198802
119894d119909
= intΩ
119880119894
119898
sum
119897=1
120597
120597119909119897
(119863119894
120597119880119894
120597119909119897
) d119909
minus 119888119894(119905) intΩ
1198802
119894119889119909 +
119899
sum
119895=1
119886119894119895(119905) intΩ
119880119894
times [119891119895(119906119895(119905 119909 120601) minus 119891
119895(119906119895(119905 119909 120595)))] d119909
+ intΩ
119880119894
[
[
119899
⋀
119895=1
120572119894119895(119905) 119891119895(119906119895(119905 minus 120591119895(119905) 119909 120601))
minus
119899
⋀
119895=1
120572119894119895(119905) 119891119895(119906119895(119905 minus 120591119895(119905) 119909 120595))]
]
d119909
+ intΩ
119880119894
[
[
119899
⋁
119895=1
120573119894119895(119905) 119891119895(119906119895(119905 minus 120591119895(119905) 119909 120601))
minus
119899
⋁
119895=1
120573119894119895(119905) 119891119895(119906119895(119905 minus 120591119895(119905) 119909 120595))]
]
d119909
(7)
for 119905 = 119905119896 119909 isin Ω 119894 = 1 119899 By boundary condition and
Green Formula we can get
intΩ
119880119894
119898
sum
119897=1
120597
120597119909119897
(119863119894
120597119880119894
120597119909119897
)119889119909 le minus119863119894intΩ
(nabla119880119894)2d119909 (8)
Then from (8) (9) (H1)-(H2) Lemma 4 and the Holderinequality
119889
119889119905
10038171003817100381710038171198801198941003817100381710038171003817
2
2
le minus2119888119894
10038171003817100381710038171198801198941003817100381710038171003817
2
2+ 2
119899
sum
119895=1
10038161003816100381610038161003816119886119894119895
10038161003816100381610038161003816119865119895
100381710038171003817100381711988011989410038171003817100381710038172
10038171003817100381710038171003817119880119895
100381710038171003817100381710038172
+ 2
119899
sum
119895=1
(10038161003816100381610038161003816120572119894119895
10038161003816100381610038161003816+10038161003816100381610038161003816120573119894119895
10038161003816100381610038161003816) 119866119895
100381710038171003817100381711988011989410038171003817100381710038172
times10038171003817100381710038171003817119906119895(119905 minus 120591119895(119905) 119909 120601) minus 119906
119895(119905 minus 120591119895(119905) 119909 120595)
100381710038171003817100381710038172
119905 = 119905119896
(9)
Thus
119863+1003817100381710038171003817119880119894
10038171003817100381710038172
le minus119888119894
100381710038171003817100381711988011989410038171003817100381710038172
+
119899
sum
119895=1
10038161003816100381610038161003816120572119894119895
10038161003816100381610038161003816119865119895
10038171003817100381710038171003817119880119895
100381710038171003817100381710038172
+
119899
sum
119895=1
(10038161003816100381610038161003816120572119894119895
10038161003816100381610038161003816+10038161003816100381610038161003816120573119894119895
10038161003816100381610038161003816) 119866119895
times10038171003817100381710038171003817119906119895(119905 minus 120591119895(119905) 119909 120601) minus 119906
119895(119905 minus 120591119895(119905) 119909 120595)
100381710038171003817100381710038172
119905 = 119905119896
(10)
for 119894 = 1 119899 Since 119862 minus (119860119865 + (120572 + 120573)119866) is a nonsingular119872-matrix there exists a vector 120585 = (120585
1 120585
119899)Tgt 0 such that
minus120585119894119888119894+
119899
sum
119895=1
120585119895(10038161003816100381610038161003816119886119894119895
10038161003816100381610038161003816119865119895+ (
10038161003816100381610038161003816120572119894119895
10038161003816100381610038161003816+10038161003816100381610038161003816120573119894119895
10038161003816100381610038161003816) 119866119895) lt 0 (11)
Considering functions
Ψ119894(119910) = 120585
119894(119910 minus 119888
119894)
+
119899
sum
119895=1
120585119895(10038161003816100381610038161003816119886119894119895
10038161003816100381610038161003816119865119895+ 119890120591119910
(10038161003816100381610038161003816120572119894119895
10038161003816100381610038161003816+10038161003816100381610038161003816120573119894119895
10038161003816100381610038161003816) 119866119895)
119894 = 1 119899
(12)
we know from (11) that Ψ119894(0) lt 0 and Ψ
119894(119910) is continuous
Since dΨ119894(119910)d119910 gt 0 Ψ
119894(119910) is strictly monotonically
increasing there exists a scalar 120582119894gt 0 such that
Ψ119894(120582119894) = 120585119894(120582119894minus 119888119894)
+
119899
sum
119895=1
120585119895(10038161003816100381610038161003816119886119894119895
10038161003816100381610038161003816119865119895+ 119890120591120582119894 (10038161003816100381610038161003816120572119894119895
10038161003816100381610038161003816+10038161003816100381610038161003816120573119894119895
10038161003816100381610038161003816) 119866119895) = 0
119894 = 1 119899
(13)
Abstract and Applied Analysis 5
Choosing 0 lt 120582 lt min1205821 120582
119899 we have
120585119894(120582119894minus 119888119894)
+
119899
sum
119895=1
120585119895(10038161003816100381610038161003816119886119894119895
10038161003816100381610038161003816119865119895+ 119890120591120582119894 (10038161003816100381610038161003816120572119894119895
10038161003816100381610038161003816+10038161003816100381610038161003816120573119894119895
10038161003816100381610038161003816) 119866119895) lt 0
119894 = 1 119899
(14)
That is
120582120585 minus (119862 minus 119860119865) 120585 + (120572 + 120573)119866120585119890minus120582119905
lt 0 (15)
Furthermore choose a positive scalar 119901 large enough suchthat
119901119890minus120582119905
120585 gt (1 1 1)T 119905 isin [minus120591 0] (16)
For any 120576 gt 0 let
119903 (119905) = 119901119890minus120582119905
(1003817100381710038171003817120601 minus 120595
10038171003817100381710038172+ 120576) 120585 119905
0le 119905 lt 119905
1 (17)
From (15)ndash(17) we obtain
119863+
119903 (119905) gt minus (119862 minus 119860119865) 119903 (119905) + (120572 + 120573)119866119903119904
(119905)
= 119881 (119905 119903 (119905) 119903119904
(119905)) 1199050le 119905 lt 119905
1
(18)
where 119903119904(119905) = (119903119904
1(119905) 119903
119904
119899(119905))
T and 119903119904
119894(119905) = sup
minus120591le119904le0119901119890minus120582(119905+119904)
(120601 minus 1205932+ 120576)120585119894 It is easy to verify that 119881(119905 119903(119905) 119903
119904
(119905)) is an119872-function It follows also from (16) and (17) that
100381710038171003817100381711988011989410038171003817100381710038172
le1003817100381710038171003817120601 minus 120593
10038171003817100381710038172lt 119901119890minus120582119905
120585119894
1003817100381710038171003817120601 minus 12059310038171003817100381710038172
lt 119903119894(119905) 119905 isin [minus120591 0] 119894 = 1 2 119899
(19)
Denote
119880
= (10038171003817100381710038171199061 (119905 119909 120601) minus 119906
1(119905 119909 120595)
10038171003817100381710038172
1003817100381710038171003817119906119899 (119905 119909 120601) minus 119906119899(119905 119909 120595)
10038171003817100381710038172)T
119880(119904)
= (10038171003817100381710038171199061 (119905 119909 120601) minus 119906
1(119905 119909 120595)
1003817100381710038171003817
(119904)
2
1003817100381710038171003817119906119899 (119905 119909 120601) minus 119906119899(119905 119909 120595)
1003817100381710038171003817
(119904)
2)
T
(20)
where 119880119894(119904)
2= sup
minus120591le119904le0119906119894(119905 + 119904 119909 120601) minus 119906
119894(119905 + 119904 119909 120595)
2
then
119880
lt 119903 (119905) 119905 isin [minus120591 0] (21)
From (10) we can obtain
119863+
119880
le minus (119862 minus 119860119865)119880
+ (120572 + 120573)119866119880(119904)
= 119881 (119905 119880
119880(119904)
) 119905 = 119905119896
(22)
Now it follows from (18)ndash(22) and Lemma 5 that
119880
lt 119903 (119905) = 119901119890minus120582119905
(1003817100381710038171003817120601 minus 120595
10038171003817100381710038172+ 120576) 120585
1199050le 119905 lt 119905
1
(23)
Letting 120576 rarr 0 we have
119880
le 1199011205851003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890minus120582119905
1199050le 119905 lt 119905
1 (24)
And moreover from (24) we get
(
119899
sum
119894=1
10038171003817100381710038171198801198941003817100381710038171003817
2
2)
12
le 119901(
119899
sum
119894=1
1205852
119894)
12
1003817100381710038171003817120601 minus 12059510038171003817100381710038172119890minus120582119905
1199050le 119905 lt 119905
1
(25)
Let = 119901(sum119899
119894=11205852
119894)12 then ge 1 Define119882(119905) = 119906
119905(119909 120601)minus
119906119905(119909 120595)
2 it follows from (25) and the definitions of 119906
119905(120601 119909)
and 119906119905(120595 119909) that
119882(119905) =1003817100381710038171003817119906119905(119909 120601) minus 119906
119905(119909 120595)
10038171003817100381710038172
le 1003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890minus120582119905
1199050le 119905 lt 119905
1
(26)
It is easily observed that
119882(119905) le 1003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890minus120582119905
minus120591 le 119905 le 1199050= 0 (27)
Because (26) holds we can suppose that for 119897 le 119896 inequality
119882(119905) le 1205780sdot sdot sdot 120578119897minus1
1003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890minus120582119905
119905119897minus1
le 119905 lt 119905119897
(28)
holds where 1205780= 1 When 119897 = 119896 + 1 we note (H5) that
119882(119905119896) =
10038171003817100381710038171003817119906119905119896
(119909 120601) minus 119906119905119896
(119909 120595)100381710038171003817100381710038172
=10038171003817100381710038171003817ℎ119896(119906minus
119905119896
(119909 120601)) minus ℎ119896(119906minus
119905119896
(119909 120595))100381710038171003817100381710038172
le 120588 (Γ2
119896)10038171003817100381710038171003817119906minus
119905119896
(119909 120601) minus 119906minus
119905119896
(119909 120595)100381710038171003817100381710038172
= 120588 (Γ2
119896)119882 (119905
minus
119896)
le 1205780sdot sdot sdot 120578119897minus1
120588 (Γ2
119896)
1003817100381710038171003817120601 minus 12059510038171003817100381710038172119890minus120582119905119896
le 1205780sdot sdot sdot 120578119897minus1
120578119896120588 (Γ2
119896)
1003817100381710038171003817120601 minus 12059510038171003817100381710038172119890minus120582119905119896
(29)
where 120588(Γ2
119896) is the spectral radius of Γ
2
119896 Let 119872 =
max 120588(Γ2
119896) by (28) (29) and 120578 ge 1 we obtain
119882(119905) le 1205780sdot sdot sdot 120578119897minus1
120578119896119872
1003817100381710038171003817120601 minus 12059510038171003817100381710038172119890minus120582119905
119905119896minus 120591 le 119905 le 119905
119896
(30)
Combining (10) (17) (30) and Lemma 5 we get
119882(119905) le 1205780sdot sdot sdot 120578119897minus1
120578119896119872
1003817100381710038171003817120601 minus 12059510038171003817100381710038172119890minus120582119905
119905119896le 119905 lt 119905
119896+1 119896 isin 119873
+
(31)
Applying mathematical induction we conclude that
119882(119905) le 1205780sdot sdot sdot 120578119897minus1
1198721003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890minus120582119905
119905119896minus1
le 119905 lt 119905119896 119896 isin 119873
+
(32)
6 Abstract and Applied Analysis
From (H6) and (32) we have
119882(119905) le 1198901205781199051119890120578(1199052minus1199051)
sdot sdot sdot 119890120578(119905119896minus1minus119905119896minus2)
times1198721003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890minus120582119905
le 1198721003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890120578119905
119890minus120582119905
= 1198721003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890minus(120582minus120578)119905
119905119896minus1
le 119905 lt 119905119896 119896 isin 119873
+
(33)
This means that1003817100381710038171003817119906119905(119909 120601) minus 119906
119905(119909 120595)
10038171003817100381710038172
le 1198721003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890minus(120582minus120578)119905
le 1198721003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890minus(120582minus120578)(119905minus120591)
119905 ge 1199050
(34)
choosing a positive integer119873 such that
119872119890minus(120582minus120578)(119873120596minus120591)
le1
6 (35)
Define a Poincare mappingD Γ rarr Γ by
D (120601) = 119906120596(119909 120601) (36)
Then
D119873
(120601) = 119906119873120596
(119909 120601) (37)
Setting 119905 = 119873120596 in (34) from (35) and (37) we have
10038171003817100381710038171003817D119873
(120601) minusD119873
(120595)100381710038171003817100381710038172
le1
6
1003817100381710038171003817120601 minus 12059510038171003817100381710038172 (38)
which implies thatD119873 is a contraction mapping Thus thereexists a unique fixed point 120601lowast isin Γ such that
D119873
(D (120601lowast
)) = D (D119873
(120601lowast
)) = D (120601lowast
) (39)
From (37) we know that D(120601lowast
) is also a fixed point of D119873and then it follows from the uniqueness of the fixed point that
D (120601lowast
) = 120601lowast
that is 119906120596(119909 120601lowast
) = 120601lowast
(40)
Let 119906(119905 119909 120601lowast
) be a solution of the model (1) then 119906(119905 +
120596 119909 120601lowast
) is also a solution of the model (1) Obviously
119906119905+120596
(119909 120601lowast
) = 119906119905(119906120596(119909 120601lowast
)) = 119906119905(119909 120601lowast
) (41)
for all 119905 ge 1199050 Hence 119906(119905 + 120596 119909 120601
lowast
) = 119906(119905 119909 120601lowast
) whichshows that 119906(119905 119909 120601lowast) is exactly one 120596-periodic solution ofmodel (1) It is easy to see that all other solutions of model (1)converge to this periodic solution exponentially as 119905 rarr +infinand the exponential convergence rate index is 120582minus120578The proofis completed
Remark 7 When 119888119894(119905) = 119888
119894 119886119894119895(119905) = 119886
119894119895 119887119894119895(119905) = 119887
119894119895 120572119894119895(119905) = 120572
119894119895
120573119894119895(119905) = 120573
119894119895 119879119894119895(119905) = 119879
119894119895119867119894119895(119905) = 119867
119894119895 119907119894(119905) = 119907
119894 119868119894(119905) = 119868
119894 and
120591119905= 120591119894(119888119894 119886119894119895 119887119894119895 120572119894119895 120573119894119895 119879119894119895 119867119894119895 119907119894 119868119894 and 120591
119894are constants)
then the model (1) is changed into
120597119906119894(119905 119909)
120597119905=
119898
sum
119897=1
120597
120597119909119897
(119863119894
120597119906119894(119905 119909)
120597119909119897
)
minus 119888119894119906119894(119905 119909) +
119899
sum
119895=1
119886119894119895119891119895(119906119895(119905 119909))
+
119899
sum
119895=1
119887119894119895119907119895+ 119869119894
+
119899
⋀
119895=1
120572119894119895119892119895(119906119895(119905 minus 120591119895(119905) 119909))
+
119899
⋁
119895=1
120573119894119895119892119895(119906119895(119905 minus 120591119895(119905) 119909))
+
119899
⋀
119895=1
119879119894119895119907119895+
119899
⋁
119895=1
119867119894119895119907119895 119905 = 119905
119896 119909 isin Ω
119906119894(119905+
119896 119909) minus 119906
119894(119905minus
119896 119909) = 119868
119894119896(119906119894(119905minus
119896 119909))
119905 = 119905119896 119896 isin Z
+ 119909 isin Ω
120597119906119894(119905 119909)
120597119899= 0 119905 ge 119905
0 119909 isin 120597Ω
119906119894(1199050+ 119904 119909) = 120595
119894(119904 119909) minus120591
119895le 119904 le 0 119909 isin Ω
(42)
For any positive constant 120596 ge 0 we have 119888119894(119905 + 120596) = 119888
119894(119905)
119886119894119895(119905 + 120596) = 119886
119894119895(119905) 119887119894119895(119905 + 120596) = 119887
119894119895(119905) 120572119894119895(119905 + 120596) = 120572
119894119895(119905)
120573119894119895(119905 + 120596) = 120573
119894119895(119905) 119879119894119895(119905 + 120596) = 119879
119894119895(119905) 119867
119894119895(119905 + 120596) = 119867
119894119895(119905)
119907119894(119905+120596) = 119907
119894(119905) 119868119894(119905+120596) = 119868
119894(119905) and 120591
119894(119905+120596) = 120591
119894(119905) for 119905 ge 119905
0
Thus the sufficient conditions inTheorem 6 are satisfied
Remark 8 If 119868119896(sdot) = 0 the model (1) is changed into
120597119906119894(119905 119909)
120597119905=
119898
sum
119897=1
120597
120597119909119897
(119863119894
120597119906119894(119905 119909)
120597119909119897
)
minus 119888119894(119905) 119906119894(119905 119909) +
119899
sum
119895=1
119886119894119895(119905) 119891119895(119906119895(119905 119909))
+
119899
sum
119895=1
119887119894119895(119905) 119907119895(119905) + 119869
119894(119905)
+
119899
⋀
119895=1
120572119894119895(119905) 119892119895(119906119895(119905 minus 120591119895(119905) 119909))
+
119899
⋁
119895=1
120573119894119895(119905) 119892119895(119906119895(119905 minus 120591119895(119905) 119909))
Abstract and Applied Analysis 7
6
4
2
0minus5
05
1015
20 76
54
32
10
1199061
119909
119905
5 43
21
Figure 1 State response 1199061(119905 119909) of model (44) without impulsiveeffects
+
119899
⋀
119895=1
119879119894119895(119905) 119907119895(119905) +
119899
⋁
119895=1
119867119894119895(119905) 119907119895(119905)
119905 = 119905119896 119909 isin Ω
120597119906119894(119905 119909)
120597119899= 0 119905 ge 119905
0 119909 isin 120597Ω
119906119894(1199050+ 119904 119909) = 120595
119894(119904 119909) minus120591
119895le 119904 le 0 119909 isin Ω
(43)
which has been discussed in [22] As Song and Wang havepointed out the model (43) is more general than some well-studied fuzzy neural networks For example when 119888
119894(119905) gt
0 119886119894119895(119905) 119887119894119895(119905) 120572119894119895(119905) 120573119894119895(119905) 119879119894119895(119905)119867119894119895(119905) 119907119894(119905) and 119868
119894(119905) are all
constants the model in (43) reduces the model which hasbeen studied by Huang [19] Moreover if 119863
119894= 0 120591
119894(119905) = 0
119891119894(120579) = 119892
119894(120579) = (12)(|120579 + 1| minus |120579 minus 1|) (119894 = 1 119899) then
model (42) covers the model studied by Yang et al [4 5] as aspecial case If119863
119894= 0 and 120591
119895(119905) is assumed to be differentiable
for 119894 119895 = 1 2 119899 then model (43) can be specialized tothe model investigated in Liu and Tang [12] and Yuan et al[13] Obviously our results are less conservative than that ofthe above-mentioned literature because they do not considerimpulsive effects
4 Numerical Examples
Example 9 Consider a two-neuron FIRDDCNNmodel
120597119906119894(119905 119909)
120597119905=
119898
sum
119897=1
120597
120597119909119897
(119863119894
120597119906119894(119905 119909)
120597119909119897
)
minus 119888119894(119905) 119906119894(119905 119909)
6
4
2
0minus5
05
1015
20 76
54
32
10
1199061
119909
119905
Figure 2 State response 1199061(119905 119909) of model (44) with impulsiveeffects
+
2
sum
119895=1
119886119894119895(119905) 119891119895(119906119895(119905 119909))
+
2
sum
119895=1
119887119894119895(119905) 119907119895(119905) + 119869
119894(119905)
+
2
⋀
119895=1
120572119894119895(119905) 119892119895(119906119895(119905 minus 120591119895(119905) 119909))
+
2
⋁
119895=1
120573119894119895(119905) 119892119895(119906119895(119905 minus 120591119895(119905) 119909))
+
2
⋀
119895=1
119879119894119895(119905) 119907119895(119905) +
2
⋁
119895=1
119867119894119895(119905) 119907119895(119905)
119905 = 119905119896 119909 isin Ω
119906119894(119905+
119896 119909) = (1 minus 120574
119894119896) 119906119894(119905minus
119896 119909)
119905 = 119905119896 119896 isin Z
+ 119909 isin Ω
120597119906119894(119905 119909)
120597119899= 0 119905 ge 119905
0 119909 isin 120597Ω
119906119894(1199050+ 119904 119909) = 120595
119894(119904 119909)
minus 120591119895le 119904 le 0 119909 isin Ω
(44)
where 119894 = 1 2 1198881(119905) = 26 119888
2(119905) = 208 119886
11(119905) = minus1 minus cos(119905)
11988612(119905) = 1 + cos(119905) 119886
21(119905) = 1 + sin(119905) 119886
22(119905) = minus1 minus sin(119905)
1198631= 8119863
2= 4 120597119906
119894(119905 119909)120597119899 = 0 (119905 ge 119905
0 119909 = 0 2120587) 120574
1119896= 04
1205742119896
= 02 1205951(sdot) = 120595
1(sdot) = 5 119887
11(119905) = 119887
21(119905) = cos(119905) 119887
12(119905) =
11988722(119905) = minus cos(119905) 119869
1(119905) = 119869
2(119905) = 1 119867
11(119905) = 119867
21(119905) = sin(119905)
11986712(119905) = 119867
22(119905) = minus1 + sin(119905) 119879
11(119905) = 119879
21(119905) = minus sin(119905)
11987912(119905) = 119879
22(119905) = 2 + sin(119905) 120591
1(119905) = 120591
2(119905) = 1 119891
119895(119906119895) =
119906119895(119905 119909) (119895 = 1 2) 119892
119895(119906119895(119905 minus 1 119909)) = 119906
119895(119905 minus 1 119909)119890
minus119906119895(119905minus1119909)
(119895 =
1 2) 12057211(119905) = minus128 120572
21(119905) = 120572
12(119905) = minus1 + cos(119905) 120572
22(119905) =
minus10 12057311(119905) = 128 120573
12(119905) = minus1 + sin(119905) = 120573
21(119905) 12057322(119905) =
8 Abstract and Applied Analysis
6
4
2
0minus5
05
1015
20 76
54
32
10
1199062
119909
119905
5 43
21
119909
Figure 3 State response 1199062(119905 119909) of model (44) without impulsiveeffects
6
4
2
0minus5
05
1015
20 76
54
32
10
1199062
119909
119905
5 43
21
119909
Figure 4 State response 1199062(119905 119909) of model (44) with impulsiveeffects
10 119907119895(119905) = sin(119905) We assume that there exists 119902 = 6 such
that 119905119896+ 2120587 = 119905
119896+119902 Obviously 119891
1 1198912 1198921 and 119892
2satisfy the
assumption (H1) with 1198651= 1198652= 1198661= 1198662= 1 and (H2) and
(H3) are satisfied with a common positive period 2120587
119862 minus (119860 + 120572 + 120573)119865 =[[
[
2
50
04
5
]]
]
(45)
is a nonsingular 119872-matrix The conditions of Theorem 6 aresatisfied hence there exists exactly one 2120596-periodic solutionof the model and all other solutions of the model convergeexponentially to it as 119905 rarr +infin Furthermore the exponentialconverging index can be calculated as120582 = 0021 because here120578119896= 1 and 120578 = 0 The simulation results are shown in Figures
1 2 3 and 4 respectively
5 Conclusions
In this paper periodicity and global exponential stability ofa class of FIRDDCNN model with variable both coefficients
and delays have been investigated By using Halanayrsquos delaydifferential inequality 119872-matrix theory and analytic meth-ods some new sufficient conditions have been established toguarantee the existence uniqueness and global exponentialstability of the periodic solution Moreover the exponentialconvergence rate index can be estimated An example andits simulation have been given to show the effectiveness ofthe obtained results In particular the differentiability ofthe time-varying delays has been removed The dynamicbehaviors of fuzzy neural networks with the property ofexponential periodicity are of great importance inmany areassuch as learning systems
Acknowledgments
The authors would like to thank the editors and the anony-mous reviewers for their valuable comments and construc-tive suggestions This research is supported by the NaturalScience Foundation of Guangxi Autonomous Region (no2012GXNSFBA053003) the National Natural Science Foun-dations of China (60973048 61272077 60974025 60673101and 60939003) National 863 Plan Project (2008 AA04Z4012009 AA043404) the Natural Science Foundation of Shan-dong Province (no Y2007G30) the Scientific andTechnolog-ical Project of Shandong Province (no 2007GG3WZ04016)the Science Foundation of Harbin Institute of Technology(Weihai) (HIT (WH) 200807) theNatural ScientificResearchInnovation Foundation in Harbin Institute of Technology(HIT NSRIF 2001120) the China Postdoctoral ScienceFoundation (2010048 1000) and the Shandong ProvincialKey Laboratory of Industrial Control Technique (QingdaoUniversity)
References
[1] LOChua andL Yang ldquoCellular neural networks theoryrdquo IEEETransactions on Circuits and Systems vol 35 no 10 pp 1257ndash1272 1988
[2] L O Chua and L Yang ldquoCellular neural networks applica-tionsrdquo IEEE Transactions on Circuits and Systems vol 35 no10 pp 1273ndash1290 1988
[3] T Roska and L O Chua ldquoCellular neural networks withnon-linear and delay-type template elements and non-uniformgridsrdquo International Journal of Circuit Theory and Applicationsvol 20 no 5 pp 469ndash481 1992
[4] T Yang L B Yang C W Wu and L O Chua ldquoFuzzycellular neural networks theoryrdquo in Proceedings of the 4th IEEEInternational Workshop on Cellular Neural Networks and TheirApplications (CNNA rsquo96) pp 181ndash186 June 1996
[5] T Yang and L-B Yang ldquoThe global stability of fuzzy cellularneural networkrdquo IEEE Transactions on Circuits and Systems Ivol 43 no 10 pp 880ndash883 1996
[6] T Yang L B Yang C W Wu and L O Chua ldquoFuzzy cellularneural networks applicationsrdquo in Proceedings of the 4th IEEEInternational Workshop on Cellular Neural Networks and TheirApplications (CNNA rsquo96) pp 225ndash230 June 1996
[7] S T Wang K F L Chung and F Duan ldquoApplying theimproved fuzzy cellular neural network IFCNN to white bloodcell detectionrdquo Neurocomputing vol 70 no 7ndash9 pp 1348ndash13592007
Abstract and Applied Analysis 9
[8] T G Barbounis and J B Theocharis ldquoA locally recurrent fuzzyneural network with application to the wind speed predictionusing spatial correlationrdquo Neurocomputing vol 70 no 7ndash9 pp1525ndash1542 2007
[9] H Zhao ldquoGlobal exponential stability and periodicity of cellularneural networks with variable delaysrdquoPhysics Letters A vol 336no 4-5 pp 331ndash341 2005
[10] S Townley A Ilchmann M G Weiss et al ldquoExistence andlearning of oscillations in recurrent neural networksrdquo IEEETransactions on Neural Networks vol 11 no 1 pp 205ndash2142000
[11] Z Huang and Y Xia ldquoExponential periodic attractor ofimpulsive BAM networks with finite distributed delaysrdquo ChaosSolitons amp Fractals vol 39 no 1 pp 373ndash384 2009
[12] Y Liu and W Tang ldquoExponential stability of fuzzy cellularneural networkswith constant and time-varying delaysrdquoPhysicsLetters A vol 323 no 3-4 pp 224ndash233 2004
[13] K Yuan J Cao and J Deng ldquoExponential stability and periodicsolutions of fuzzy cellular neural networks with time-varyingdelaysrdquoNeurocomputing vol 69 no 13ndash15 pp 1619ndash1627 2006
[14] T Huang ldquoExponential stability of fuzzy cellular neural net-works with distributed delayrdquo Physics Letters A vol 351 no 1-2pp 48ndash52 2006
[15] G Bao S Wen and Z Zeng ldquoRobust stability analysis of inter-val fuzzy Cohen-Grossberg neural networks with piecewiseconstant argument of generalized typerdquo Neural Networks vol33 pp 32ndash41 2012
[16] H Zhao N Ding and L Chen ldquoAlmost sure exponential sta-bility of stochastic fuzzy cellular neural networks with delaysrdquoChaos Solitons amp Fractals vol 40 no 4 pp 1653ndash1659 2009
[17] W Ding M Han and M Li ldquoExponential lag synchronizationof delayed fuzzy cellular neural networkswith impulsesrdquoPhysicsLetters A vol 373 no 8-9 pp 832ndash837 2009
[18] L Chen and H Zhao ldquoStability analysis of stochastic fuzzycellular neural networks with delaysrdquo Neurocomputing vol 72no 1ndash3 pp 436ndash444 2008
[19] T Huang ldquoExponential stability of delayed fuzzy cellular neuralnetworks with diffusionrdquo Chaos Solitons amp Fractals vol 31 no3 pp 658ndash664 2007
[20] J Wang and J G Lu ldquoGlobal exponential stability of fuzzycellular neural networks with delays and reaction-diffusiontermsrdquo Chaos Solitons amp Fractals vol 38 no 3 pp 878ndash8852008
[21] Q Song and Z Wang ldquoDynamical behaviors of fuzzy reaction-diffusion periodic cellular neural networks with variable coeffi-cients and delaysrdquo Applied Mathematical Modelling vol 33 no9 pp 3533ndash3545 2009
[22] C Wang Y kao and G Yang ldquoExponential stability ofimpulsive stochastic fuzzy reaction-diffusion Cohen-Grossbergneural networks with mixed delaysrdquo Neurocomputing vol 89pp 55ndash63 2012
[23] T Serrano-Gotarredona and B Linares-Barranco ldquoLog-Domain Implementation of Complex Dynamics Reaction-Diffusion Neural Networksrdquo IEEE Transactions on NeuralNetworks vol 14 no 5 pp 1337ndash1355 2003
[24] J Liang and J Cao ldquoGlobal exponential stability of reaction-diffusion recurrent neural networks with time-varying delaysrdquoPhysics Letters A vol 314 no 5-6 pp 434ndash442 2003
[25] L Wang and D Xu ldquoAsymptotic behavior of a class of reaction-diffusion equations with delaysrdquo Journal of Mathematical Anal-ysis and Applications vol 281 no 2 pp 439ndash453 2003
[26] L Wang and D Xu ldquoGlobal exponential stability of Hopfieldreaction-diffusion neural networks with time-varying delaysrdquoScience in China Series F vol 46 no 6 pp 466ndash474 2003
[27] Z Yang and D Xu ldquoGlobal dynamics for non-autonomousreaction-diffusion neural networks with time-varying delaysrdquoTheoretical Computer Science vol 403 no 1 pp 3ndash10 2008
[28] Y G Kao C C Gao and D K Wang ldquoGlobal exponentialstability of reaction-diffusion Hopfield neural networks withcontinuously distributed delaysrdquo Mathematica Applicata vol21 no 3 pp 457ndash462 2008 (Chinese)
[29] Y-G Kao J-F Guo C-H Wang and X-Q Sun ldquoDelay-dependent robust exponential stability of Markovian jump-ing reaction-diffusion Cohen-Grossberg neural networks withmixed delaysrdquo Journal of the Franklin Institute vol 349 no 6pp 1972ndash1988 2012
[30] H Zhao and Z Mao ldquoBoundedness and stability of nonau-tonomous cellular neural networks with reaction-diffusiontermsrdquo Mathematics and Computers in Simulation vol 79 no5 pp 1603ndash1617 2009
[31] H Zhao and G Wang ldquoExistence of periodic oscillatorysolution of reaction-diffusion neural networks with delaysrdquoPhysics Letters A vol 343 no 5 pp 372ndash383 2005
[32] Q Gan ldquoExponential synchronization of stochastic Cohen-Grossberg neural networks withmixed time-varying delays andreaction-diffusion via periodically intermittent controlrdquoNeuralNetworks vol 31 pp 12ndash21 2012
[33] V Lakshmikantham D D Baınov and P S Simeonov Theoryof Impulsive Differential Equations vol 6 of Series in ModernApplied Mathematics World Scientific Teaneck NJ USA 1989
[34] D D Bainov and P S Simeonov Impulsive Differential Equa-tions Periodic Solutions and Applications Longman HarlowUK 1993
[35] X Song X Xin andWHuang ldquoExponential stability of delayedand impulsive cellular neural networks with partially Lipschitzcontinuous activation functionsrdquo Neural Networks vol 29-30pp 80ndash90 2012
[36] Z Yang and D Xu ldquoExistence and exponential stability ofperiodic solution for impulsive delay differential equations andapplicationsrdquo Nonlinear Analysis Theory Methods amp Applica-tions vol 64 no 1 pp 130ndash145 2006
[37] X Yang X Liao D J Evans and Y Tang ldquoExistence andstability of periodic solution in impulsive Hopfield neuralnetworks with finite distributed delaysrdquo Physics Letters A vol343 no 1ndash3 pp 108ndash116 2005
[38] Y Li and L Lu ldquoGlobal exponential stability and existenceof periodic solution of Hopfield-type neural networks withimpulsesrdquo Physics Letters A vol 333 no 1-2 pp 62ndash71 2004
[39] Y Li L Zhu andP Liu ldquoExistence and stability of periodic solu-tions of delayed cellular neural networksrdquo Nonlinear AnalysisReal World Applications vol 7 no 2 pp 225ndash234 2006
[40] Z H Guan and G Chen ldquoOn delayed impulsive Hopfieldneural networksrdquo Neural Networks vol 12 no 2 pp 273ndash2801999
[41] Z H Guan J Lam andG Chen ldquoOn impulsive autoassociativeneural networksrdquo Neural Networks vol 13 no 1 pp 63ndash692000
[42] W Ding ldquoSynchronization of delayed fuzzy cellular neuralnetworks with impulsive effectsrdquo Communications in NonlinearScience and Numerical Simulation vol 14 no 11 pp 3945ndash39522009
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Abstract and Applied Analysis 3
and introduce the norm 120595(119905)2= (sum119899
119894=1120595119894(119905)2
2)12 where
120595119894(119905)2= (intΩ
|120595119894(119905 119909)|
2
119889119909)12
Throughout the paper we make the following assump-tions
(H1) There exists a positive diagonal matrix 119865 =
diag(1198651 1198652 119865
119899) and 119866 = diag(119866
1 1198662 119866
119899)
such that
119865119895= sup119909 = 119910
1003816100381610038161003816100381610038161003816100381610038161003816
119891119895(119909) minus 119891
119895(119910)
119909 minus 119910
1003816100381610038161003816100381610038161003816100381610038161003816
119866119895= sup119909 = 119910
1003816100381610038161003816100381610038161003816100381610038161003816
119892119895(119909) minus 119892
119895(119910)
119909 minus 119910
1003816100381610038161003816100381610038161003816100381610038161003816
(2)
for all 119909 = 119910 119895 = 1 2 119899
(H2) 119888119894(119905) gt 0 119886
119894119895(119905) 119887119894119895(119905) 120572119894119895(119905) 120573119894119895(119905) 119879119894119895(119905)119867119894119895(119905) 119907119894(119905)
119868119894(119905) and 120591
119895(119905) ge 0 are periodic function with a
common positive period 120596 for all 119905 ge 1199050 119894 119895 = 1
2 119899
(H3) For120596 gt 0 119894 = 1 2 119899 there exists 119902 isin 119885+such that
119905119896+ 120596 = 119905
119896+119902 119868119894119896(119906119894) = 119868119894(119896+119902)
(119906119894) and 119868
119894119896(119906119894(119905119896 119909)) are
Lipschitz continuous in R119899
Definition 1 Themodel in (1) is said to be globally exponen-tially periodic if (i) there exists one 120596-periodic solution and(ii) all other solutions of the model converge exponentially toit as 119905 rarr +infin
Definition 2 (see [26]) Let C = ([119905 minus 120591 119905]R119899) where 120591 ge
0 and 119865(119905 119909 119910) isin C(R+ times R119899 times CR119899) Then the function119865(119905 119909 119910) = (119891
1(119905 119909 119910) 119891
2(119905 119909 119910) 119891
119899(119905 119909 119910))
T is calledan 119872-function if (i) for every 119905 isin R+ 119909 isin R119899 119910(1) isin Cthere holds 119865(119905 119909 119910(1)) le 119865(119905 119909 119910
(2)
) for 119910(1)
le 119910(2) where
119910(1)
= (119910(1)
1 119910
(1)
119899)T and 119910
(2)
= (119910(2)
1 119910
(2)
119899)T (ii) every
119894th element of 119865 satisfies 119891119894(119905 119909(1)
119910) le 119891119894(119905 119909(2)
119910) for any119910 isin C 119905 ge 119905
0 where arbitrary 119909
(1) and 119909(2)
(119909(1)
le 119909(2)
)
belong to R119899 and have the same 119894th component 119909(1)119894
= 119909(2)
119894
Here 119909(1) = (119909(1)
1 119909
(1)
119899)T 119909(2)
= (119909(2)
1 119909
(2)
119899)T
Definition 3 (see [26]) A real matrix119860 = (119886119894119895)119899times119899
is said to bea nonsingular 119872-matrix if 119886
119894119895le 0 (119894 = 119895 119894 119895 = 1 119899) and
all successive principal minors of 119860 are positive
Lemma 4 (see [13]) Let 119906 and 119906lowast be two states of the model
in (1) then we have
10038161003816100381610038161003816100381610038161003816100381610038161003816
119899
⋀
119895=1
120572119894119895(119905) 119891119895(119906119895) minus
119899
⋀
119895=1
120572119894119895(119905) 119891119895(119906lowast
119895)
10038161003816100381610038161003816100381610038161003816100381610038161003816
le
119899
sum
119895=1
10038161003816100381610038161003816120572119894119895(119905)
10038161003816100381610038161003816sdot10038161003816100381610038161003816119891119895(119906119895) minus 119891119895(119906lowast
119895)10038161003816100381610038161003816
10038161003816100381610038161003816100381610038161003816100381610038161003816
119899
⋁
119895=1
120573119894119895(119905) 119891119895(119906119895) minus
119899
⋁
119895=1
120573119894119895(119905) 119891119895(119906lowast
119895)
10038161003816100381610038161003816100381610038161003816100381610038161003816
le
119899
sum
119895=1
10038161003816100381610038161003816120573119894119895(119905)
10038161003816100381610038161003816sdot10038161003816100381610038161003816119891119895(119906119895) minus 119891119895(119906lowast
119895)10038161003816100381610038161003816
(3)
Lemma 5 (see [26]) Assume that 119865(119905 119909 119910) is an119872-functionand (i) 119909(119905) lt 119910(119905) 119905 isin [119905 minus 120591 119905
0] (ii) 119863
+
119910(119905) gt
119865(119905 119910(119905) 119910119904
(119905)) 119863+119909(119905) le 119865(119905 119909(119905) 119909119904
(119905)) 119905 ge 1199050 where
119909119904
(119905) = supminus120591le119904le0
119909(119905 + 119904) 119910119904(119905) = supminus120591le119904le0
119910(119905 + 119904) Then119909(119905) lt 119910(119905) 119905 ge 119905
0
3 Main Results and Proofs
We should first point out that under assumptions (H1)(H2) and (H3) the FIRDDCNN model (1) has at least one120596-periodic solution of [26] The proof of the existence ofthe 120596-periodic solution of (1) can be carried out similar to[26 28] by the nonlinear functional analysis methods such astopological degree andhere is omittedWewillmainly discussthe uniqueness of the periodic solution and its exponentialstability
Theorem 6 Assume that (H1)ndash(H3) holds Furthermoreassume that the following conditions hold
(H4) C minus 119860F minus (120572 + 120573)119866 is a nonsingular119872-matrix
(H5) The impulsive operators ℎ119896(119906) = 119906 + 119868
119896(119906) is Lipschitz
continuous in R119899 that is there exists a nonnegativediagnose matrix Γ
119896= diag(1205741k 120574nk) such that
|ℎ119896(119906) minus ℎ
119896(119906lowast
)| le Γ119896|119906 minus 119906
lowast
| for all 119906 119906lowast isin R119899119896 isin 119873
+ where |ℎ119896(119906)| = (|ℎ
1119896(1199061)| |ℎ
119899119896(119906119899)|)
T119868119896(119906) = (119868
1119896(1199061) 119868
119899119896(119906119899))T
(H6) 120578 = sup119896isin119873+ln 120578119896(119905119896minus 119905119896minus1
) lt 120582 where 120578119896
=
max1le119894le119899
1 120574119894119896 119896 isin 119873
+
Then the model (1) is global exponential periodic andthe exponential convergence rate index 120582 minus 120578 and 120582 can beestimated by
120585119894(120582 minus 119888
119894)
+
119899
sum
119895=1
120585119895(10038161003816100381610038161003816119886119894119895
10038161003816100381610038161003816119865119895+ 119890120591120582
(10038161003816100381610038161003816120572119894119895
10038161003816100381610038161003816+10038161003816100381610038161003816120573119894119895
10038161003816100381610038161003816) 119866119895) lt 0
119894 = 1 119899
(4)
where 119862 = diag(1198881 119888
119899) and 120585
119894gt 0 119860 = (|119886
119894119895|)119899times119899
120572 =
(|120572119894119895|)119899times119899
120573 = (|120573119894119895|)119899times119899
satisfies minus120585119894119888119894+sum119899
119895=1120585119894(|119886119894119895|119865119894+
(|120572119894119895| + |120573119894119895|)119866119894) lt 0
Proof For any 120601 120595 isin PCΩ let 119906(119905 119909 120601) = (119906
1(119905 119909 120601)
119906119899(119905 119909 120601))
T be a periodic solution of the system (1) starting
4 Abstract and Applied Analysis
from 120601 and 119906(119905 119909 120595) = (1199061(119905 119909 120595) 119906
119899(119905 119909 120595))
T a solu-tion of the system (1) starting from 120595 Define
119906119905(120601 119909) = 119906 (119905 + 119904 119909 120601)
119906119905(120595 119909) = 119906 (119905 + 119904 119909 120595) 119904 isin [minus120591 0]
(5)
and we can see that 119906119905(120601 119909) 119906
119905(120595 119909) isin PC
Ωfor all 119905 gt 0 Let
119880119894= 119906119894(119905 119909 120601) minus 119906
119894(119905 119909 120595) then from (1) we get
120597119880119894
120597119905=
119898
sum
119897=1
120597
120597119909119897
(119863119894
120597119880119894
120597119909119897
) minus 119888119894(119905) 119880119894+
119899
sum
119895=1
119886119894119895(119905)
times [119891119895(119906119895(119905 119909 120601)) minus 119891
119895(119906119895(119905 119909 120595))]
+ [
[
119899
⋀
119895=1
120572119894119895(119905) 119891119895(119906119895(119905 minus 120591119895(119905) 119909 120601))
minus
119899
⋀
119895=1
120572119894119895(119905) 119891119895(119906119895(119905 minus 120591119895(119905) 119909 120595))]
]
+ [
[
119899
⋁
119895=1
120573119894119895(119905) 119891119895(119906119895(119905 minus 120591119895(119905) 119909 120601))
minus
119899
⋁
119895=1
120573119894119895(119905) 119891119895(119906119895(119905 minus 120591119895(119905) 119909 120595))]
]
(6)
for all 119905 = 119905119896 119909 isin Ω 119894 = 1 119899
Multiplying both sides of (6) by 119880119894and integrating it in
Ω we have1
2
dd119905
intΩ
1198802
119894d119909
= intΩ
119880119894
119898
sum
119897=1
120597
120597119909119897
(119863119894
120597119880119894
120597119909119897
) d119909
minus 119888119894(119905) intΩ
1198802
119894119889119909 +
119899
sum
119895=1
119886119894119895(119905) intΩ
119880119894
times [119891119895(119906119895(119905 119909 120601) minus 119891
119895(119906119895(119905 119909 120595)))] d119909
+ intΩ
119880119894
[
[
119899
⋀
119895=1
120572119894119895(119905) 119891119895(119906119895(119905 minus 120591119895(119905) 119909 120601))
minus
119899
⋀
119895=1
120572119894119895(119905) 119891119895(119906119895(119905 minus 120591119895(119905) 119909 120595))]
]
d119909
+ intΩ
119880119894
[
[
119899
⋁
119895=1
120573119894119895(119905) 119891119895(119906119895(119905 minus 120591119895(119905) 119909 120601))
minus
119899
⋁
119895=1
120573119894119895(119905) 119891119895(119906119895(119905 minus 120591119895(119905) 119909 120595))]
]
d119909
(7)
for 119905 = 119905119896 119909 isin Ω 119894 = 1 119899 By boundary condition and
Green Formula we can get
intΩ
119880119894
119898
sum
119897=1
120597
120597119909119897
(119863119894
120597119880119894
120597119909119897
)119889119909 le minus119863119894intΩ
(nabla119880119894)2d119909 (8)
Then from (8) (9) (H1)-(H2) Lemma 4 and the Holderinequality
119889
119889119905
10038171003817100381710038171198801198941003817100381710038171003817
2
2
le minus2119888119894
10038171003817100381710038171198801198941003817100381710038171003817
2
2+ 2
119899
sum
119895=1
10038161003816100381610038161003816119886119894119895
10038161003816100381610038161003816119865119895
100381710038171003817100381711988011989410038171003817100381710038172
10038171003817100381710038171003817119880119895
100381710038171003817100381710038172
+ 2
119899
sum
119895=1
(10038161003816100381610038161003816120572119894119895
10038161003816100381610038161003816+10038161003816100381610038161003816120573119894119895
10038161003816100381610038161003816) 119866119895
100381710038171003817100381711988011989410038171003817100381710038172
times10038171003817100381710038171003817119906119895(119905 minus 120591119895(119905) 119909 120601) minus 119906
119895(119905 minus 120591119895(119905) 119909 120595)
100381710038171003817100381710038172
119905 = 119905119896
(9)
Thus
119863+1003817100381710038171003817119880119894
10038171003817100381710038172
le minus119888119894
100381710038171003817100381711988011989410038171003817100381710038172
+
119899
sum
119895=1
10038161003816100381610038161003816120572119894119895
10038161003816100381610038161003816119865119895
10038171003817100381710038171003817119880119895
100381710038171003817100381710038172
+
119899
sum
119895=1
(10038161003816100381610038161003816120572119894119895
10038161003816100381610038161003816+10038161003816100381610038161003816120573119894119895
10038161003816100381610038161003816) 119866119895
times10038171003817100381710038171003817119906119895(119905 minus 120591119895(119905) 119909 120601) minus 119906
119895(119905 minus 120591119895(119905) 119909 120595)
100381710038171003817100381710038172
119905 = 119905119896
(10)
for 119894 = 1 119899 Since 119862 minus (119860119865 + (120572 + 120573)119866) is a nonsingular119872-matrix there exists a vector 120585 = (120585
1 120585
119899)Tgt 0 such that
minus120585119894119888119894+
119899
sum
119895=1
120585119895(10038161003816100381610038161003816119886119894119895
10038161003816100381610038161003816119865119895+ (
10038161003816100381610038161003816120572119894119895
10038161003816100381610038161003816+10038161003816100381610038161003816120573119894119895
10038161003816100381610038161003816) 119866119895) lt 0 (11)
Considering functions
Ψ119894(119910) = 120585
119894(119910 minus 119888
119894)
+
119899
sum
119895=1
120585119895(10038161003816100381610038161003816119886119894119895
10038161003816100381610038161003816119865119895+ 119890120591119910
(10038161003816100381610038161003816120572119894119895
10038161003816100381610038161003816+10038161003816100381610038161003816120573119894119895
10038161003816100381610038161003816) 119866119895)
119894 = 1 119899
(12)
we know from (11) that Ψ119894(0) lt 0 and Ψ
119894(119910) is continuous
Since dΨ119894(119910)d119910 gt 0 Ψ
119894(119910) is strictly monotonically
increasing there exists a scalar 120582119894gt 0 such that
Ψ119894(120582119894) = 120585119894(120582119894minus 119888119894)
+
119899
sum
119895=1
120585119895(10038161003816100381610038161003816119886119894119895
10038161003816100381610038161003816119865119895+ 119890120591120582119894 (10038161003816100381610038161003816120572119894119895
10038161003816100381610038161003816+10038161003816100381610038161003816120573119894119895
10038161003816100381610038161003816) 119866119895) = 0
119894 = 1 119899
(13)
Abstract and Applied Analysis 5
Choosing 0 lt 120582 lt min1205821 120582
119899 we have
120585119894(120582119894minus 119888119894)
+
119899
sum
119895=1
120585119895(10038161003816100381610038161003816119886119894119895
10038161003816100381610038161003816119865119895+ 119890120591120582119894 (10038161003816100381610038161003816120572119894119895
10038161003816100381610038161003816+10038161003816100381610038161003816120573119894119895
10038161003816100381610038161003816) 119866119895) lt 0
119894 = 1 119899
(14)
That is
120582120585 minus (119862 minus 119860119865) 120585 + (120572 + 120573)119866120585119890minus120582119905
lt 0 (15)
Furthermore choose a positive scalar 119901 large enough suchthat
119901119890minus120582119905
120585 gt (1 1 1)T 119905 isin [minus120591 0] (16)
For any 120576 gt 0 let
119903 (119905) = 119901119890minus120582119905
(1003817100381710038171003817120601 minus 120595
10038171003817100381710038172+ 120576) 120585 119905
0le 119905 lt 119905
1 (17)
From (15)ndash(17) we obtain
119863+
119903 (119905) gt minus (119862 minus 119860119865) 119903 (119905) + (120572 + 120573)119866119903119904
(119905)
= 119881 (119905 119903 (119905) 119903119904
(119905)) 1199050le 119905 lt 119905
1
(18)
where 119903119904(119905) = (119903119904
1(119905) 119903
119904
119899(119905))
T and 119903119904
119894(119905) = sup
minus120591le119904le0119901119890minus120582(119905+119904)
(120601 minus 1205932+ 120576)120585119894 It is easy to verify that 119881(119905 119903(119905) 119903
119904
(119905)) is an119872-function It follows also from (16) and (17) that
100381710038171003817100381711988011989410038171003817100381710038172
le1003817100381710038171003817120601 minus 120593
10038171003817100381710038172lt 119901119890minus120582119905
120585119894
1003817100381710038171003817120601 minus 12059310038171003817100381710038172
lt 119903119894(119905) 119905 isin [minus120591 0] 119894 = 1 2 119899
(19)
Denote
119880
= (10038171003817100381710038171199061 (119905 119909 120601) minus 119906
1(119905 119909 120595)
10038171003817100381710038172
1003817100381710038171003817119906119899 (119905 119909 120601) minus 119906119899(119905 119909 120595)
10038171003817100381710038172)T
119880(119904)
= (10038171003817100381710038171199061 (119905 119909 120601) minus 119906
1(119905 119909 120595)
1003817100381710038171003817
(119904)
2
1003817100381710038171003817119906119899 (119905 119909 120601) minus 119906119899(119905 119909 120595)
1003817100381710038171003817
(119904)
2)
T
(20)
where 119880119894(119904)
2= sup
minus120591le119904le0119906119894(119905 + 119904 119909 120601) minus 119906
119894(119905 + 119904 119909 120595)
2
then
119880
lt 119903 (119905) 119905 isin [minus120591 0] (21)
From (10) we can obtain
119863+
119880
le minus (119862 minus 119860119865)119880
+ (120572 + 120573)119866119880(119904)
= 119881 (119905 119880
119880(119904)
) 119905 = 119905119896
(22)
Now it follows from (18)ndash(22) and Lemma 5 that
119880
lt 119903 (119905) = 119901119890minus120582119905
(1003817100381710038171003817120601 minus 120595
10038171003817100381710038172+ 120576) 120585
1199050le 119905 lt 119905
1
(23)
Letting 120576 rarr 0 we have
119880
le 1199011205851003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890minus120582119905
1199050le 119905 lt 119905
1 (24)
And moreover from (24) we get
(
119899
sum
119894=1
10038171003817100381710038171198801198941003817100381710038171003817
2
2)
12
le 119901(
119899
sum
119894=1
1205852
119894)
12
1003817100381710038171003817120601 minus 12059510038171003817100381710038172119890minus120582119905
1199050le 119905 lt 119905
1
(25)
Let = 119901(sum119899
119894=11205852
119894)12 then ge 1 Define119882(119905) = 119906
119905(119909 120601)minus
119906119905(119909 120595)
2 it follows from (25) and the definitions of 119906
119905(120601 119909)
and 119906119905(120595 119909) that
119882(119905) =1003817100381710038171003817119906119905(119909 120601) minus 119906
119905(119909 120595)
10038171003817100381710038172
le 1003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890minus120582119905
1199050le 119905 lt 119905
1
(26)
It is easily observed that
119882(119905) le 1003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890minus120582119905
minus120591 le 119905 le 1199050= 0 (27)
Because (26) holds we can suppose that for 119897 le 119896 inequality
119882(119905) le 1205780sdot sdot sdot 120578119897minus1
1003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890minus120582119905
119905119897minus1
le 119905 lt 119905119897
(28)
holds where 1205780= 1 When 119897 = 119896 + 1 we note (H5) that
119882(119905119896) =
10038171003817100381710038171003817119906119905119896
(119909 120601) minus 119906119905119896
(119909 120595)100381710038171003817100381710038172
=10038171003817100381710038171003817ℎ119896(119906minus
119905119896
(119909 120601)) minus ℎ119896(119906minus
119905119896
(119909 120595))100381710038171003817100381710038172
le 120588 (Γ2
119896)10038171003817100381710038171003817119906minus
119905119896
(119909 120601) minus 119906minus
119905119896
(119909 120595)100381710038171003817100381710038172
= 120588 (Γ2
119896)119882 (119905
minus
119896)
le 1205780sdot sdot sdot 120578119897minus1
120588 (Γ2
119896)
1003817100381710038171003817120601 minus 12059510038171003817100381710038172119890minus120582119905119896
le 1205780sdot sdot sdot 120578119897minus1
120578119896120588 (Γ2
119896)
1003817100381710038171003817120601 minus 12059510038171003817100381710038172119890minus120582119905119896
(29)
where 120588(Γ2
119896) is the spectral radius of Γ
2
119896 Let 119872 =
max 120588(Γ2
119896) by (28) (29) and 120578 ge 1 we obtain
119882(119905) le 1205780sdot sdot sdot 120578119897minus1
120578119896119872
1003817100381710038171003817120601 minus 12059510038171003817100381710038172119890minus120582119905
119905119896minus 120591 le 119905 le 119905
119896
(30)
Combining (10) (17) (30) and Lemma 5 we get
119882(119905) le 1205780sdot sdot sdot 120578119897minus1
120578119896119872
1003817100381710038171003817120601 minus 12059510038171003817100381710038172119890minus120582119905
119905119896le 119905 lt 119905
119896+1 119896 isin 119873
+
(31)
Applying mathematical induction we conclude that
119882(119905) le 1205780sdot sdot sdot 120578119897minus1
1198721003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890minus120582119905
119905119896minus1
le 119905 lt 119905119896 119896 isin 119873
+
(32)
6 Abstract and Applied Analysis
From (H6) and (32) we have
119882(119905) le 1198901205781199051119890120578(1199052minus1199051)
sdot sdot sdot 119890120578(119905119896minus1minus119905119896minus2)
times1198721003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890minus120582119905
le 1198721003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890120578119905
119890minus120582119905
= 1198721003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890minus(120582minus120578)119905
119905119896minus1
le 119905 lt 119905119896 119896 isin 119873
+
(33)
This means that1003817100381710038171003817119906119905(119909 120601) minus 119906
119905(119909 120595)
10038171003817100381710038172
le 1198721003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890minus(120582minus120578)119905
le 1198721003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890minus(120582minus120578)(119905minus120591)
119905 ge 1199050
(34)
choosing a positive integer119873 such that
119872119890minus(120582minus120578)(119873120596minus120591)
le1
6 (35)
Define a Poincare mappingD Γ rarr Γ by
D (120601) = 119906120596(119909 120601) (36)
Then
D119873
(120601) = 119906119873120596
(119909 120601) (37)
Setting 119905 = 119873120596 in (34) from (35) and (37) we have
10038171003817100381710038171003817D119873
(120601) minusD119873
(120595)100381710038171003817100381710038172
le1
6
1003817100381710038171003817120601 minus 12059510038171003817100381710038172 (38)
which implies thatD119873 is a contraction mapping Thus thereexists a unique fixed point 120601lowast isin Γ such that
D119873
(D (120601lowast
)) = D (D119873
(120601lowast
)) = D (120601lowast
) (39)
From (37) we know that D(120601lowast
) is also a fixed point of D119873and then it follows from the uniqueness of the fixed point that
D (120601lowast
) = 120601lowast
that is 119906120596(119909 120601lowast
) = 120601lowast
(40)
Let 119906(119905 119909 120601lowast
) be a solution of the model (1) then 119906(119905 +
120596 119909 120601lowast
) is also a solution of the model (1) Obviously
119906119905+120596
(119909 120601lowast
) = 119906119905(119906120596(119909 120601lowast
)) = 119906119905(119909 120601lowast
) (41)
for all 119905 ge 1199050 Hence 119906(119905 + 120596 119909 120601
lowast
) = 119906(119905 119909 120601lowast
) whichshows that 119906(119905 119909 120601lowast) is exactly one 120596-periodic solution ofmodel (1) It is easy to see that all other solutions of model (1)converge to this periodic solution exponentially as 119905 rarr +infinand the exponential convergence rate index is 120582minus120578The proofis completed
Remark 7 When 119888119894(119905) = 119888
119894 119886119894119895(119905) = 119886
119894119895 119887119894119895(119905) = 119887
119894119895 120572119894119895(119905) = 120572
119894119895
120573119894119895(119905) = 120573
119894119895 119879119894119895(119905) = 119879
119894119895119867119894119895(119905) = 119867
119894119895 119907119894(119905) = 119907
119894 119868119894(119905) = 119868
119894 and
120591119905= 120591119894(119888119894 119886119894119895 119887119894119895 120572119894119895 120573119894119895 119879119894119895 119867119894119895 119907119894 119868119894 and 120591
119894are constants)
then the model (1) is changed into
120597119906119894(119905 119909)
120597119905=
119898
sum
119897=1
120597
120597119909119897
(119863119894
120597119906119894(119905 119909)
120597119909119897
)
minus 119888119894119906119894(119905 119909) +
119899
sum
119895=1
119886119894119895119891119895(119906119895(119905 119909))
+
119899
sum
119895=1
119887119894119895119907119895+ 119869119894
+
119899
⋀
119895=1
120572119894119895119892119895(119906119895(119905 minus 120591119895(119905) 119909))
+
119899
⋁
119895=1
120573119894119895119892119895(119906119895(119905 minus 120591119895(119905) 119909))
+
119899
⋀
119895=1
119879119894119895119907119895+
119899
⋁
119895=1
119867119894119895119907119895 119905 = 119905
119896 119909 isin Ω
119906119894(119905+
119896 119909) minus 119906
119894(119905minus
119896 119909) = 119868
119894119896(119906119894(119905minus
119896 119909))
119905 = 119905119896 119896 isin Z
+ 119909 isin Ω
120597119906119894(119905 119909)
120597119899= 0 119905 ge 119905
0 119909 isin 120597Ω
119906119894(1199050+ 119904 119909) = 120595
119894(119904 119909) minus120591
119895le 119904 le 0 119909 isin Ω
(42)
For any positive constant 120596 ge 0 we have 119888119894(119905 + 120596) = 119888
119894(119905)
119886119894119895(119905 + 120596) = 119886
119894119895(119905) 119887119894119895(119905 + 120596) = 119887
119894119895(119905) 120572119894119895(119905 + 120596) = 120572
119894119895(119905)
120573119894119895(119905 + 120596) = 120573
119894119895(119905) 119879119894119895(119905 + 120596) = 119879
119894119895(119905) 119867
119894119895(119905 + 120596) = 119867
119894119895(119905)
119907119894(119905+120596) = 119907
119894(119905) 119868119894(119905+120596) = 119868
119894(119905) and 120591
119894(119905+120596) = 120591
119894(119905) for 119905 ge 119905
0
Thus the sufficient conditions inTheorem 6 are satisfied
Remark 8 If 119868119896(sdot) = 0 the model (1) is changed into
120597119906119894(119905 119909)
120597119905=
119898
sum
119897=1
120597
120597119909119897
(119863119894
120597119906119894(119905 119909)
120597119909119897
)
minus 119888119894(119905) 119906119894(119905 119909) +
119899
sum
119895=1
119886119894119895(119905) 119891119895(119906119895(119905 119909))
+
119899
sum
119895=1
119887119894119895(119905) 119907119895(119905) + 119869
119894(119905)
+
119899
⋀
119895=1
120572119894119895(119905) 119892119895(119906119895(119905 minus 120591119895(119905) 119909))
+
119899
⋁
119895=1
120573119894119895(119905) 119892119895(119906119895(119905 minus 120591119895(119905) 119909))
Abstract and Applied Analysis 7
6
4
2
0minus5
05
1015
20 76
54
32
10
1199061
119909
119905
5 43
21
Figure 1 State response 1199061(119905 119909) of model (44) without impulsiveeffects
+
119899
⋀
119895=1
119879119894119895(119905) 119907119895(119905) +
119899
⋁
119895=1
119867119894119895(119905) 119907119895(119905)
119905 = 119905119896 119909 isin Ω
120597119906119894(119905 119909)
120597119899= 0 119905 ge 119905
0 119909 isin 120597Ω
119906119894(1199050+ 119904 119909) = 120595
119894(119904 119909) minus120591
119895le 119904 le 0 119909 isin Ω
(43)
which has been discussed in [22] As Song and Wang havepointed out the model (43) is more general than some well-studied fuzzy neural networks For example when 119888
119894(119905) gt
0 119886119894119895(119905) 119887119894119895(119905) 120572119894119895(119905) 120573119894119895(119905) 119879119894119895(119905)119867119894119895(119905) 119907119894(119905) and 119868
119894(119905) are all
constants the model in (43) reduces the model which hasbeen studied by Huang [19] Moreover if 119863
119894= 0 120591
119894(119905) = 0
119891119894(120579) = 119892
119894(120579) = (12)(|120579 + 1| minus |120579 minus 1|) (119894 = 1 119899) then
model (42) covers the model studied by Yang et al [4 5] as aspecial case If119863
119894= 0 and 120591
119895(119905) is assumed to be differentiable
for 119894 119895 = 1 2 119899 then model (43) can be specialized tothe model investigated in Liu and Tang [12] and Yuan et al[13] Obviously our results are less conservative than that ofthe above-mentioned literature because they do not considerimpulsive effects
4 Numerical Examples
Example 9 Consider a two-neuron FIRDDCNNmodel
120597119906119894(119905 119909)
120597119905=
119898
sum
119897=1
120597
120597119909119897
(119863119894
120597119906119894(119905 119909)
120597119909119897
)
minus 119888119894(119905) 119906119894(119905 119909)
6
4
2
0minus5
05
1015
20 76
54
32
10
1199061
119909
119905
Figure 2 State response 1199061(119905 119909) of model (44) with impulsiveeffects
+
2
sum
119895=1
119886119894119895(119905) 119891119895(119906119895(119905 119909))
+
2
sum
119895=1
119887119894119895(119905) 119907119895(119905) + 119869
119894(119905)
+
2
⋀
119895=1
120572119894119895(119905) 119892119895(119906119895(119905 minus 120591119895(119905) 119909))
+
2
⋁
119895=1
120573119894119895(119905) 119892119895(119906119895(119905 minus 120591119895(119905) 119909))
+
2
⋀
119895=1
119879119894119895(119905) 119907119895(119905) +
2
⋁
119895=1
119867119894119895(119905) 119907119895(119905)
119905 = 119905119896 119909 isin Ω
119906119894(119905+
119896 119909) = (1 minus 120574
119894119896) 119906119894(119905minus
119896 119909)
119905 = 119905119896 119896 isin Z
+ 119909 isin Ω
120597119906119894(119905 119909)
120597119899= 0 119905 ge 119905
0 119909 isin 120597Ω
119906119894(1199050+ 119904 119909) = 120595
119894(119904 119909)
minus 120591119895le 119904 le 0 119909 isin Ω
(44)
where 119894 = 1 2 1198881(119905) = 26 119888
2(119905) = 208 119886
11(119905) = minus1 minus cos(119905)
11988612(119905) = 1 + cos(119905) 119886
21(119905) = 1 + sin(119905) 119886
22(119905) = minus1 minus sin(119905)
1198631= 8119863
2= 4 120597119906
119894(119905 119909)120597119899 = 0 (119905 ge 119905
0 119909 = 0 2120587) 120574
1119896= 04
1205742119896
= 02 1205951(sdot) = 120595
1(sdot) = 5 119887
11(119905) = 119887
21(119905) = cos(119905) 119887
12(119905) =
11988722(119905) = minus cos(119905) 119869
1(119905) = 119869
2(119905) = 1 119867
11(119905) = 119867
21(119905) = sin(119905)
11986712(119905) = 119867
22(119905) = minus1 + sin(119905) 119879
11(119905) = 119879
21(119905) = minus sin(119905)
11987912(119905) = 119879
22(119905) = 2 + sin(119905) 120591
1(119905) = 120591
2(119905) = 1 119891
119895(119906119895) =
119906119895(119905 119909) (119895 = 1 2) 119892
119895(119906119895(119905 minus 1 119909)) = 119906
119895(119905 minus 1 119909)119890
minus119906119895(119905minus1119909)
(119895 =
1 2) 12057211(119905) = minus128 120572
21(119905) = 120572
12(119905) = minus1 + cos(119905) 120572
22(119905) =
minus10 12057311(119905) = 128 120573
12(119905) = minus1 + sin(119905) = 120573
21(119905) 12057322(119905) =
8 Abstract and Applied Analysis
6
4
2
0minus5
05
1015
20 76
54
32
10
1199062
119909
119905
5 43
21
119909
Figure 3 State response 1199062(119905 119909) of model (44) without impulsiveeffects
6
4
2
0minus5
05
1015
20 76
54
32
10
1199062
119909
119905
5 43
21
119909
Figure 4 State response 1199062(119905 119909) of model (44) with impulsiveeffects
10 119907119895(119905) = sin(119905) We assume that there exists 119902 = 6 such
that 119905119896+ 2120587 = 119905
119896+119902 Obviously 119891
1 1198912 1198921 and 119892
2satisfy the
assumption (H1) with 1198651= 1198652= 1198661= 1198662= 1 and (H2) and
(H3) are satisfied with a common positive period 2120587
119862 minus (119860 + 120572 + 120573)119865 =[[
[
2
50
04
5
]]
]
(45)
is a nonsingular 119872-matrix The conditions of Theorem 6 aresatisfied hence there exists exactly one 2120596-periodic solutionof the model and all other solutions of the model convergeexponentially to it as 119905 rarr +infin Furthermore the exponentialconverging index can be calculated as120582 = 0021 because here120578119896= 1 and 120578 = 0 The simulation results are shown in Figures
1 2 3 and 4 respectively
5 Conclusions
In this paper periodicity and global exponential stability ofa class of FIRDDCNN model with variable both coefficients
and delays have been investigated By using Halanayrsquos delaydifferential inequality 119872-matrix theory and analytic meth-ods some new sufficient conditions have been established toguarantee the existence uniqueness and global exponentialstability of the periodic solution Moreover the exponentialconvergence rate index can be estimated An example andits simulation have been given to show the effectiveness ofthe obtained results In particular the differentiability ofthe time-varying delays has been removed The dynamicbehaviors of fuzzy neural networks with the property ofexponential periodicity are of great importance inmany areassuch as learning systems
Acknowledgments
The authors would like to thank the editors and the anony-mous reviewers for their valuable comments and construc-tive suggestions This research is supported by the NaturalScience Foundation of Guangxi Autonomous Region (no2012GXNSFBA053003) the National Natural Science Foun-dations of China (60973048 61272077 60974025 60673101and 60939003) National 863 Plan Project (2008 AA04Z4012009 AA043404) the Natural Science Foundation of Shan-dong Province (no Y2007G30) the Scientific andTechnolog-ical Project of Shandong Province (no 2007GG3WZ04016)the Science Foundation of Harbin Institute of Technology(Weihai) (HIT (WH) 200807) theNatural ScientificResearchInnovation Foundation in Harbin Institute of Technology(HIT NSRIF 2001120) the China Postdoctoral ScienceFoundation (2010048 1000) and the Shandong ProvincialKey Laboratory of Industrial Control Technique (QingdaoUniversity)
References
[1] LOChua andL Yang ldquoCellular neural networks theoryrdquo IEEETransactions on Circuits and Systems vol 35 no 10 pp 1257ndash1272 1988
[2] L O Chua and L Yang ldquoCellular neural networks applica-tionsrdquo IEEE Transactions on Circuits and Systems vol 35 no10 pp 1273ndash1290 1988
[3] T Roska and L O Chua ldquoCellular neural networks withnon-linear and delay-type template elements and non-uniformgridsrdquo International Journal of Circuit Theory and Applicationsvol 20 no 5 pp 469ndash481 1992
[4] T Yang L B Yang C W Wu and L O Chua ldquoFuzzycellular neural networks theoryrdquo in Proceedings of the 4th IEEEInternational Workshop on Cellular Neural Networks and TheirApplications (CNNA rsquo96) pp 181ndash186 June 1996
[5] T Yang and L-B Yang ldquoThe global stability of fuzzy cellularneural networkrdquo IEEE Transactions on Circuits and Systems Ivol 43 no 10 pp 880ndash883 1996
[6] T Yang L B Yang C W Wu and L O Chua ldquoFuzzy cellularneural networks applicationsrdquo in Proceedings of the 4th IEEEInternational Workshop on Cellular Neural Networks and TheirApplications (CNNA rsquo96) pp 225ndash230 June 1996
[7] S T Wang K F L Chung and F Duan ldquoApplying theimproved fuzzy cellular neural network IFCNN to white bloodcell detectionrdquo Neurocomputing vol 70 no 7ndash9 pp 1348ndash13592007
Abstract and Applied Analysis 9
[8] T G Barbounis and J B Theocharis ldquoA locally recurrent fuzzyneural network with application to the wind speed predictionusing spatial correlationrdquo Neurocomputing vol 70 no 7ndash9 pp1525ndash1542 2007
[9] H Zhao ldquoGlobal exponential stability and periodicity of cellularneural networks with variable delaysrdquoPhysics Letters A vol 336no 4-5 pp 331ndash341 2005
[10] S Townley A Ilchmann M G Weiss et al ldquoExistence andlearning of oscillations in recurrent neural networksrdquo IEEETransactions on Neural Networks vol 11 no 1 pp 205ndash2142000
[11] Z Huang and Y Xia ldquoExponential periodic attractor ofimpulsive BAM networks with finite distributed delaysrdquo ChaosSolitons amp Fractals vol 39 no 1 pp 373ndash384 2009
[12] Y Liu and W Tang ldquoExponential stability of fuzzy cellularneural networkswith constant and time-varying delaysrdquoPhysicsLetters A vol 323 no 3-4 pp 224ndash233 2004
[13] K Yuan J Cao and J Deng ldquoExponential stability and periodicsolutions of fuzzy cellular neural networks with time-varyingdelaysrdquoNeurocomputing vol 69 no 13ndash15 pp 1619ndash1627 2006
[14] T Huang ldquoExponential stability of fuzzy cellular neural net-works with distributed delayrdquo Physics Letters A vol 351 no 1-2pp 48ndash52 2006
[15] G Bao S Wen and Z Zeng ldquoRobust stability analysis of inter-val fuzzy Cohen-Grossberg neural networks with piecewiseconstant argument of generalized typerdquo Neural Networks vol33 pp 32ndash41 2012
[16] H Zhao N Ding and L Chen ldquoAlmost sure exponential sta-bility of stochastic fuzzy cellular neural networks with delaysrdquoChaos Solitons amp Fractals vol 40 no 4 pp 1653ndash1659 2009
[17] W Ding M Han and M Li ldquoExponential lag synchronizationof delayed fuzzy cellular neural networkswith impulsesrdquoPhysicsLetters A vol 373 no 8-9 pp 832ndash837 2009
[18] L Chen and H Zhao ldquoStability analysis of stochastic fuzzycellular neural networks with delaysrdquo Neurocomputing vol 72no 1ndash3 pp 436ndash444 2008
[19] T Huang ldquoExponential stability of delayed fuzzy cellular neuralnetworks with diffusionrdquo Chaos Solitons amp Fractals vol 31 no3 pp 658ndash664 2007
[20] J Wang and J G Lu ldquoGlobal exponential stability of fuzzycellular neural networks with delays and reaction-diffusiontermsrdquo Chaos Solitons amp Fractals vol 38 no 3 pp 878ndash8852008
[21] Q Song and Z Wang ldquoDynamical behaviors of fuzzy reaction-diffusion periodic cellular neural networks with variable coeffi-cients and delaysrdquo Applied Mathematical Modelling vol 33 no9 pp 3533ndash3545 2009
[22] C Wang Y kao and G Yang ldquoExponential stability ofimpulsive stochastic fuzzy reaction-diffusion Cohen-Grossbergneural networks with mixed delaysrdquo Neurocomputing vol 89pp 55ndash63 2012
[23] T Serrano-Gotarredona and B Linares-Barranco ldquoLog-Domain Implementation of Complex Dynamics Reaction-Diffusion Neural Networksrdquo IEEE Transactions on NeuralNetworks vol 14 no 5 pp 1337ndash1355 2003
[24] J Liang and J Cao ldquoGlobal exponential stability of reaction-diffusion recurrent neural networks with time-varying delaysrdquoPhysics Letters A vol 314 no 5-6 pp 434ndash442 2003
[25] L Wang and D Xu ldquoAsymptotic behavior of a class of reaction-diffusion equations with delaysrdquo Journal of Mathematical Anal-ysis and Applications vol 281 no 2 pp 439ndash453 2003
[26] L Wang and D Xu ldquoGlobal exponential stability of Hopfieldreaction-diffusion neural networks with time-varying delaysrdquoScience in China Series F vol 46 no 6 pp 466ndash474 2003
[27] Z Yang and D Xu ldquoGlobal dynamics for non-autonomousreaction-diffusion neural networks with time-varying delaysrdquoTheoretical Computer Science vol 403 no 1 pp 3ndash10 2008
[28] Y G Kao C C Gao and D K Wang ldquoGlobal exponentialstability of reaction-diffusion Hopfield neural networks withcontinuously distributed delaysrdquo Mathematica Applicata vol21 no 3 pp 457ndash462 2008 (Chinese)
[29] Y-G Kao J-F Guo C-H Wang and X-Q Sun ldquoDelay-dependent robust exponential stability of Markovian jump-ing reaction-diffusion Cohen-Grossberg neural networks withmixed delaysrdquo Journal of the Franklin Institute vol 349 no 6pp 1972ndash1988 2012
[30] H Zhao and Z Mao ldquoBoundedness and stability of nonau-tonomous cellular neural networks with reaction-diffusiontermsrdquo Mathematics and Computers in Simulation vol 79 no5 pp 1603ndash1617 2009
[31] H Zhao and G Wang ldquoExistence of periodic oscillatorysolution of reaction-diffusion neural networks with delaysrdquoPhysics Letters A vol 343 no 5 pp 372ndash383 2005
[32] Q Gan ldquoExponential synchronization of stochastic Cohen-Grossberg neural networks withmixed time-varying delays andreaction-diffusion via periodically intermittent controlrdquoNeuralNetworks vol 31 pp 12ndash21 2012
[33] V Lakshmikantham D D Baınov and P S Simeonov Theoryof Impulsive Differential Equations vol 6 of Series in ModernApplied Mathematics World Scientific Teaneck NJ USA 1989
[34] D D Bainov and P S Simeonov Impulsive Differential Equa-tions Periodic Solutions and Applications Longman HarlowUK 1993
[35] X Song X Xin andWHuang ldquoExponential stability of delayedand impulsive cellular neural networks with partially Lipschitzcontinuous activation functionsrdquo Neural Networks vol 29-30pp 80ndash90 2012
[36] Z Yang and D Xu ldquoExistence and exponential stability ofperiodic solution for impulsive delay differential equations andapplicationsrdquo Nonlinear Analysis Theory Methods amp Applica-tions vol 64 no 1 pp 130ndash145 2006
[37] X Yang X Liao D J Evans and Y Tang ldquoExistence andstability of periodic solution in impulsive Hopfield neuralnetworks with finite distributed delaysrdquo Physics Letters A vol343 no 1ndash3 pp 108ndash116 2005
[38] Y Li and L Lu ldquoGlobal exponential stability and existenceof periodic solution of Hopfield-type neural networks withimpulsesrdquo Physics Letters A vol 333 no 1-2 pp 62ndash71 2004
[39] Y Li L Zhu andP Liu ldquoExistence and stability of periodic solu-tions of delayed cellular neural networksrdquo Nonlinear AnalysisReal World Applications vol 7 no 2 pp 225ndash234 2006
[40] Z H Guan and G Chen ldquoOn delayed impulsive Hopfieldneural networksrdquo Neural Networks vol 12 no 2 pp 273ndash2801999
[41] Z H Guan J Lam andG Chen ldquoOn impulsive autoassociativeneural networksrdquo Neural Networks vol 13 no 1 pp 63ndash692000
[42] W Ding ldquoSynchronization of delayed fuzzy cellular neuralnetworks with impulsive effectsrdquo Communications in NonlinearScience and Numerical Simulation vol 14 no 11 pp 3945ndash39522009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Stochastic AnalysisInternational Journal of
4 Abstract and Applied Analysis
from 120601 and 119906(119905 119909 120595) = (1199061(119905 119909 120595) 119906
119899(119905 119909 120595))
T a solu-tion of the system (1) starting from 120595 Define
119906119905(120601 119909) = 119906 (119905 + 119904 119909 120601)
119906119905(120595 119909) = 119906 (119905 + 119904 119909 120595) 119904 isin [minus120591 0]
(5)
and we can see that 119906119905(120601 119909) 119906
119905(120595 119909) isin PC
Ωfor all 119905 gt 0 Let
119880119894= 119906119894(119905 119909 120601) minus 119906
119894(119905 119909 120595) then from (1) we get
120597119880119894
120597119905=
119898
sum
119897=1
120597
120597119909119897
(119863119894
120597119880119894
120597119909119897
) minus 119888119894(119905) 119880119894+
119899
sum
119895=1
119886119894119895(119905)
times [119891119895(119906119895(119905 119909 120601)) minus 119891
119895(119906119895(119905 119909 120595))]
+ [
[
119899
⋀
119895=1
120572119894119895(119905) 119891119895(119906119895(119905 minus 120591119895(119905) 119909 120601))
minus
119899
⋀
119895=1
120572119894119895(119905) 119891119895(119906119895(119905 minus 120591119895(119905) 119909 120595))]
]
+ [
[
119899
⋁
119895=1
120573119894119895(119905) 119891119895(119906119895(119905 minus 120591119895(119905) 119909 120601))
minus
119899
⋁
119895=1
120573119894119895(119905) 119891119895(119906119895(119905 minus 120591119895(119905) 119909 120595))]
]
(6)
for all 119905 = 119905119896 119909 isin Ω 119894 = 1 119899
Multiplying both sides of (6) by 119880119894and integrating it in
Ω we have1
2
dd119905
intΩ
1198802
119894d119909
= intΩ
119880119894
119898
sum
119897=1
120597
120597119909119897
(119863119894
120597119880119894
120597119909119897
) d119909
minus 119888119894(119905) intΩ
1198802
119894119889119909 +
119899
sum
119895=1
119886119894119895(119905) intΩ
119880119894
times [119891119895(119906119895(119905 119909 120601) minus 119891
119895(119906119895(119905 119909 120595)))] d119909
+ intΩ
119880119894
[
[
119899
⋀
119895=1
120572119894119895(119905) 119891119895(119906119895(119905 minus 120591119895(119905) 119909 120601))
minus
119899
⋀
119895=1
120572119894119895(119905) 119891119895(119906119895(119905 minus 120591119895(119905) 119909 120595))]
]
d119909
+ intΩ
119880119894
[
[
119899
⋁
119895=1
120573119894119895(119905) 119891119895(119906119895(119905 minus 120591119895(119905) 119909 120601))
minus
119899
⋁
119895=1
120573119894119895(119905) 119891119895(119906119895(119905 minus 120591119895(119905) 119909 120595))]
]
d119909
(7)
for 119905 = 119905119896 119909 isin Ω 119894 = 1 119899 By boundary condition and
Green Formula we can get
intΩ
119880119894
119898
sum
119897=1
120597
120597119909119897
(119863119894
120597119880119894
120597119909119897
)119889119909 le minus119863119894intΩ
(nabla119880119894)2d119909 (8)
Then from (8) (9) (H1)-(H2) Lemma 4 and the Holderinequality
119889
119889119905
10038171003817100381710038171198801198941003817100381710038171003817
2
2
le minus2119888119894
10038171003817100381710038171198801198941003817100381710038171003817
2
2+ 2
119899
sum
119895=1
10038161003816100381610038161003816119886119894119895
10038161003816100381610038161003816119865119895
100381710038171003817100381711988011989410038171003817100381710038172
10038171003817100381710038171003817119880119895
100381710038171003817100381710038172
+ 2
119899
sum
119895=1
(10038161003816100381610038161003816120572119894119895
10038161003816100381610038161003816+10038161003816100381610038161003816120573119894119895
10038161003816100381610038161003816) 119866119895
100381710038171003817100381711988011989410038171003817100381710038172
times10038171003817100381710038171003817119906119895(119905 minus 120591119895(119905) 119909 120601) minus 119906
119895(119905 minus 120591119895(119905) 119909 120595)
100381710038171003817100381710038172
119905 = 119905119896
(9)
Thus
119863+1003817100381710038171003817119880119894
10038171003817100381710038172
le minus119888119894
100381710038171003817100381711988011989410038171003817100381710038172
+
119899
sum
119895=1
10038161003816100381610038161003816120572119894119895
10038161003816100381610038161003816119865119895
10038171003817100381710038171003817119880119895
100381710038171003817100381710038172
+
119899
sum
119895=1
(10038161003816100381610038161003816120572119894119895
10038161003816100381610038161003816+10038161003816100381610038161003816120573119894119895
10038161003816100381610038161003816) 119866119895
times10038171003817100381710038171003817119906119895(119905 minus 120591119895(119905) 119909 120601) minus 119906
119895(119905 minus 120591119895(119905) 119909 120595)
100381710038171003817100381710038172
119905 = 119905119896
(10)
for 119894 = 1 119899 Since 119862 minus (119860119865 + (120572 + 120573)119866) is a nonsingular119872-matrix there exists a vector 120585 = (120585
1 120585
119899)Tgt 0 such that
minus120585119894119888119894+
119899
sum
119895=1
120585119895(10038161003816100381610038161003816119886119894119895
10038161003816100381610038161003816119865119895+ (
10038161003816100381610038161003816120572119894119895
10038161003816100381610038161003816+10038161003816100381610038161003816120573119894119895
10038161003816100381610038161003816) 119866119895) lt 0 (11)
Considering functions
Ψ119894(119910) = 120585
119894(119910 minus 119888
119894)
+
119899
sum
119895=1
120585119895(10038161003816100381610038161003816119886119894119895
10038161003816100381610038161003816119865119895+ 119890120591119910
(10038161003816100381610038161003816120572119894119895
10038161003816100381610038161003816+10038161003816100381610038161003816120573119894119895
10038161003816100381610038161003816) 119866119895)
119894 = 1 119899
(12)
we know from (11) that Ψ119894(0) lt 0 and Ψ
119894(119910) is continuous
Since dΨ119894(119910)d119910 gt 0 Ψ
119894(119910) is strictly monotonically
increasing there exists a scalar 120582119894gt 0 such that
Ψ119894(120582119894) = 120585119894(120582119894minus 119888119894)
+
119899
sum
119895=1
120585119895(10038161003816100381610038161003816119886119894119895
10038161003816100381610038161003816119865119895+ 119890120591120582119894 (10038161003816100381610038161003816120572119894119895
10038161003816100381610038161003816+10038161003816100381610038161003816120573119894119895
10038161003816100381610038161003816) 119866119895) = 0
119894 = 1 119899
(13)
Abstract and Applied Analysis 5
Choosing 0 lt 120582 lt min1205821 120582
119899 we have
120585119894(120582119894minus 119888119894)
+
119899
sum
119895=1
120585119895(10038161003816100381610038161003816119886119894119895
10038161003816100381610038161003816119865119895+ 119890120591120582119894 (10038161003816100381610038161003816120572119894119895
10038161003816100381610038161003816+10038161003816100381610038161003816120573119894119895
10038161003816100381610038161003816) 119866119895) lt 0
119894 = 1 119899
(14)
That is
120582120585 minus (119862 minus 119860119865) 120585 + (120572 + 120573)119866120585119890minus120582119905
lt 0 (15)
Furthermore choose a positive scalar 119901 large enough suchthat
119901119890minus120582119905
120585 gt (1 1 1)T 119905 isin [minus120591 0] (16)
For any 120576 gt 0 let
119903 (119905) = 119901119890minus120582119905
(1003817100381710038171003817120601 minus 120595
10038171003817100381710038172+ 120576) 120585 119905
0le 119905 lt 119905
1 (17)
From (15)ndash(17) we obtain
119863+
119903 (119905) gt minus (119862 minus 119860119865) 119903 (119905) + (120572 + 120573)119866119903119904
(119905)
= 119881 (119905 119903 (119905) 119903119904
(119905)) 1199050le 119905 lt 119905
1
(18)
where 119903119904(119905) = (119903119904
1(119905) 119903
119904
119899(119905))
T and 119903119904
119894(119905) = sup
minus120591le119904le0119901119890minus120582(119905+119904)
(120601 minus 1205932+ 120576)120585119894 It is easy to verify that 119881(119905 119903(119905) 119903
119904
(119905)) is an119872-function It follows also from (16) and (17) that
100381710038171003817100381711988011989410038171003817100381710038172
le1003817100381710038171003817120601 minus 120593
10038171003817100381710038172lt 119901119890minus120582119905
120585119894
1003817100381710038171003817120601 minus 12059310038171003817100381710038172
lt 119903119894(119905) 119905 isin [minus120591 0] 119894 = 1 2 119899
(19)
Denote
119880
= (10038171003817100381710038171199061 (119905 119909 120601) minus 119906
1(119905 119909 120595)
10038171003817100381710038172
1003817100381710038171003817119906119899 (119905 119909 120601) minus 119906119899(119905 119909 120595)
10038171003817100381710038172)T
119880(119904)
= (10038171003817100381710038171199061 (119905 119909 120601) minus 119906
1(119905 119909 120595)
1003817100381710038171003817
(119904)
2
1003817100381710038171003817119906119899 (119905 119909 120601) minus 119906119899(119905 119909 120595)
1003817100381710038171003817
(119904)
2)
T
(20)
where 119880119894(119904)
2= sup
minus120591le119904le0119906119894(119905 + 119904 119909 120601) minus 119906
119894(119905 + 119904 119909 120595)
2
then
119880
lt 119903 (119905) 119905 isin [minus120591 0] (21)
From (10) we can obtain
119863+
119880
le minus (119862 minus 119860119865)119880
+ (120572 + 120573)119866119880(119904)
= 119881 (119905 119880
119880(119904)
) 119905 = 119905119896
(22)
Now it follows from (18)ndash(22) and Lemma 5 that
119880
lt 119903 (119905) = 119901119890minus120582119905
(1003817100381710038171003817120601 minus 120595
10038171003817100381710038172+ 120576) 120585
1199050le 119905 lt 119905
1
(23)
Letting 120576 rarr 0 we have
119880
le 1199011205851003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890minus120582119905
1199050le 119905 lt 119905
1 (24)
And moreover from (24) we get
(
119899
sum
119894=1
10038171003817100381710038171198801198941003817100381710038171003817
2
2)
12
le 119901(
119899
sum
119894=1
1205852
119894)
12
1003817100381710038171003817120601 minus 12059510038171003817100381710038172119890minus120582119905
1199050le 119905 lt 119905
1
(25)
Let = 119901(sum119899
119894=11205852
119894)12 then ge 1 Define119882(119905) = 119906
119905(119909 120601)minus
119906119905(119909 120595)
2 it follows from (25) and the definitions of 119906
119905(120601 119909)
and 119906119905(120595 119909) that
119882(119905) =1003817100381710038171003817119906119905(119909 120601) minus 119906
119905(119909 120595)
10038171003817100381710038172
le 1003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890minus120582119905
1199050le 119905 lt 119905
1
(26)
It is easily observed that
119882(119905) le 1003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890minus120582119905
minus120591 le 119905 le 1199050= 0 (27)
Because (26) holds we can suppose that for 119897 le 119896 inequality
119882(119905) le 1205780sdot sdot sdot 120578119897minus1
1003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890minus120582119905
119905119897minus1
le 119905 lt 119905119897
(28)
holds where 1205780= 1 When 119897 = 119896 + 1 we note (H5) that
119882(119905119896) =
10038171003817100381710038171003817119906119905119896
(119909 120601) minus 119906119905119896
(119909 120595)100381710038171003817100381710038172
=10038171003817100381710038171003817ℎ119896(119906minus
119905119896
(119909 120601)) minus ℎ119896(119906minus
119905119896
(119909 120595))100381710038171003817100381710038172
le 120588 (Γ2
119896)10038171003817100381710038171003817119906minus
119905119896
(119909 120601) minus 119906minus
119905119896
(119909 120595)100381710038171003817100381710038172
= 120588 (Γ2
119896)119882 (119905
minus
119896)
le 1205780sdot sdot sdot 120578119897minus1
120588 (Γ2
119896)
1003817100381710038171003817120601 minus 12059510038171003817100381710038172119890minus120582119905119896
le 1205780sdot sdot sdot 120578119897minus1
120578119896120588 (Γ2
119896)
1003817100381710038171003817120601 minus 12059510038171003817100381710038172119890minus120582119905119896
(29)
where 120588(Γ2
119896) is the spectral radius of Γ
2
119896 Let 119872 =
max 120588(Γ2
119896) by (28) (29) and 120578 ge 1 we obtain
119882(119905) le 1205780sdot sdot sdot 120578119897minus1
120578119896119872
1003817100381710038171003817120601 minus 12059510038171003817100381710038172119890minus120582119905
119905119896minus 120591 le 119905 le 119905
119896
(30)
Combining (10) (17) (30) and Lemma 5 we get
119882(119905) le 1205780sdot sdot sdot 120578119897minus1
120578119896119872
1003817100381710038171003817120601 minus 12059510038171003817100381710038172119890minus120582119905
119905119896le 119905 lt 119905
119896+1 119896 isin 119873
+
(31)
Applying mathematical induction we conclude that
119882(119905) le 1205780sdot sdot sdot 120578119897minus1
1198721003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890minus120582119905
119905119896minus1
le 119905 lt 119905119896 119896 isin 119873
+
(32)
6 Abstract and Applied Analysis
From (H6) and (32) we have
119882(119905) le 1198901205781199051119890120578(1199052minus1199051)
sdot sdot sdot 119890120578(119905119896minus1minus119905119896minus2)
times1198721003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890minus120582119905
le 1198721003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890120578119905
119890minus120582119905
= 1198721003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890minus(120582minus120578)119905
119905119896minus1
le 119905 lt 119905119896 119896 isin 119873
+
(33)
This means that1003817100381710038171003817119906119905(119909 120601) minus 119906
119905(119909 120595)
10038171003817100381710038172
le 1198721003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890minus(120582minus120578)119905
le 1198721003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890minus(120582minus120578)(119905minus120591)
119905 ge 1199050
(34)
choosing a positive integer119873 such that
119872119890minus(120582minus120578)(119873120596minus120591)
le1
6 (35)
Define a Poincare mappingD Γ rarr Γ by
D (120601) = 119906120596(119909 120601) (36)
Then
D119873
(120601) = 119906119873120596
(119909 120601) (37)
Setting 119905 = 119873120596 in (34) from (35) and (37) we have
10038171003817100381710038171003817D119873
(120601) minusD119873
(120595)100381710038171003817100381710038172
le1
6
1003817100381710038171003817120601 minus 12059510038171003817100381710038172 (38)
which implies thatD119873 is a contraction mapping Thus thereexists a unique fixed point 120601lowast isin Γ such that
D119873
(D (120601lowast
)) = D (D119873
(120601lowast
)) = D (120601lowast
) (39)
From (37) we know that D(120601lowast
) is also a fixed point of D119873and then it follows from the uniqueness of the fixed point that
D (120601lowast
) = 120601lowast
that is 119906120596(119909 120601lowast
) = 120601lowast
(40)
Let 119906(119905 119909 120601lowast
) be a solution of the model (1) then 119906(119905 +
120596 119909 120601lowast
) is also a solution of the model (1) Obviously
119906119905+120596
(119909 120601lowast
) = 119906119905(119906120596(119909 120601lowast
)) = 119906119905(119909 120601lowast
) (41)
for all 119905 ge 1199050 Hence 119906(119905 + 120596 119909 120601
lowast
) = 119906(119905 119909 120601lowast
) whichshows that 119906(119905 119909 120601lowast) is exactly one 120596-periodic solution ofmodel (1) It is easy to see that all other solutions of model (1)converge to this periodic solution exponentially as 119905 rarr +infinand the exponential convergence rate index is 120582minus120578The proofis completed
Remark 7 When 119888119894(119905) = 119888
119894 119886119894119895(119905) = 119886
119894119895 119887119894119895(119905) = 119887
119894119895 120572119894119895(119905) = 120572
119894119895
120573119894119895(119905) = 120573
119894119895 119879119894119895(119905) = 119879
119894119895119867119894119895(119905) = 119867
119894119895 119907119894(119905) = 119907
119894 119868119894(119905) = 119868
119894 and
120591119905= 120591119894(119888119894 119886119894119895 119887119894119895 120572119894119895 120573119894119895 119879119894119895 119867119894119895 119907119894 119868119894 and 120591
119894are constants)
then the model (1) is changed into
120597119906119894(119905 119909)
120597119905=
119898
sum
119897=1
120597
120597119909119897
(119863119894
120597119906119894(119905 119909)
120597119909119897
)
minus 119888119894119906119894(119905 119909) +
119899
sum
119895=1
119886119894119895119891119895(119906119895(119905 119909))
+
119899
sum
119895=1
119887119894119895119907119895+ 119869119894
+
119899
⋀
119895=1
120572119894119895119892119895(119906119895(119905 minus 120591119895(119905) 119909))
+
119899
⋁
119895=1
120573119894119895119892119895(119906119895(119905 minus 120591119895(119905) 119909))
+
119899
⋀
119895=1
119879119894119895119907119895+
119899
⋁
119895=1
119867119894119895119907119895 119905 = 119905
119896 119909 isin Ω
119906119894(119905+
119896 119909) minus 119906
119894(119905minus
119896 119909) = 119868
119894119896(119906119894(119905minus
119896 119909))
119905 = 119905119896 119896 isin Z
+ 119909 isin Ω
120597119906119894(119905 119909)
120597119899= 0 119905 ge 119905
0 119909 isin 120597Ω
119906119894(1199050+ 119904 119909) = 120595
119894(119904 119909) minus120591
119895le 119904 le 0 119909 isin Ω
(42)
For any positive constant 120596 ge 0 we have 119888119894(119905 + 120596) = 119888
119894(119905)
119886119894119895(119905 + 120596) = 119886
119894119895(119905) 119887119894119895(119905 + 120596) = 119887
119894119895(119905) 120572119894119895(119905 + 120596) = 120572
119894119895(119905)
120573119894119895(119905 + 120596) = 120573
119894119895(119905) 119879119894119895(119905 + 120596) = 119879
119894119895(119905) 119867
119894119895(119905 + 120596) = 119867
119894119895(119905)
119907119894(119905+120596) = 119907
119894(119905) 119868119894(119905+120596) = 119868
119894(119905) and 120591
119894(119905+120596) = 120591
119894(119905) for 119905 ge 119905
0
Thus the sufficient conditions inTheorem 6 are satisfied
Remark 8 If 119868119896(sdot) = 0 the model (1) is changed into
120597119906119894(119905 119909)
120597119905=
119898
sum
119897=1
120597
120597119909119897
(119863119894
120597119906119894(119905 119909)
120597119909119897
)
minus 119888119894(119905) 119906119894(119905 119909) +
119899
sum
119895=1
119886119894119895(119905) 119891119895(119906119895(119905 119909))
+
119899
sum
119895=1
119887119894119895(119905) 119907119895(119905) + 119869
119894(119905)
+
119899
⋀
119895=1
120572119894119895(119905) 119892119895(119906119895(119905 minus 120591119895(119905) 119909))
+
119899
⋁
119895=1
120573119894119895(119905) 119892119895(119906119895(119905 minus 120591119895(119905) 119909))
Abstract and Applied Analysis 7
6
4
2
0minus5
05
1015
20 76
54
32
10
1199061
119909
119905
5 43
21
Figure 1 State response 1199061(119905 119909) of model (44) without impulsiveeffects
+
119899
⋀
119895=1
119879119894119895(119905) 119907119895(119905) +
119899
⋁
119895=1
119867119894119895(119905) 119907119895(119905)
119905 = 119905119896 119909 isin Ω
120597119906119894(119905 119909)
120597119899= 0 119905 ge 119905
0 119909 isin 120597Ω
119906119894(1199050+ 119904 119909) = 120595
119894(119904 119909) minus120591
119895le 119904 le 0 119909 isin Ω
(43)
which has been discussed in [22] As Song and Wang havepointed out the model (43) is more general than some well-studied fuzzy neural networks For example when 119888
119894(119905) gt
0 119886119894119895(119905) 119887119894119895(119905) 120572119894119895(119905) 120573119894119895(119905) 119879119894119895(119905)119867119894119895(119905) 119907119894(119905) and 119868
119894(119905) are all
constants the model in (43) reduces the model which hasbeen studied by Huang [19] Moreover if 119863
119894= 0 120591
119894(119905) = 0
119891119894(120579) = 119892
119894(120579) = (12)(|120579 + 1| minus |120579 minus 1|) (119894 = 1 119899) then
model (42) covers the model studied by Yang et al [4 5] as aspecial case If119863
119894= 0 and 120591
119895(119905) is assumed to be differentiable
for 119894 119895 = 1 2 119899 then model (43) can be specialized tothe model investigated in Liu and Tang [12] and Yuan et al[13] Obviously our results are less conservative than that ofthe above-mentioned literature because they do not considerimpulsive effects
4 Numerical Examples
Example 9 Consider a two-neuron FIRDDCNNmodel
120597119906119894(119905 119909)
120597119905=
119898
sum
119897=1
120597
120597119909119897
(119863119894
120597119906119894(119905 119909)
120597119909119897
)
minus 119888119894(119905) 119906119894(119905 119909)
6
4
2
0minus5
05
1015
20 76
54
32
10
1199061
119909
119905
Figure 2 State response 1199061(119905 119909) of model (44) with impulsiveeffects
+
2
sum
119895=1
119886119894119895(119905) 119891119895(119906119895(119905 119909))
+
2
sum
119895=1
119887119894119895(119905) 119907119895(119905) + 119869
119894(119905)
+
2
⋀
119895=1
120572119894119895(119905) 119892119895(119906119895(119905 minus 120591119895(119905) 119909))
+
2
⋁
119895=1
120573119894119895(119905) 119892119895(119906119895(119905 minus 120591119895(119905) 119909))
+
2
⋀
119895=1
119879119894119895(119905) 119907119895(119905) +
2
⋁
119895=1
119867119894119895(119905) 119907119895(119905)
119905 = 119905119896 119909 isin Ω
119906119894(119905+
119896 119909) = (1 minus 120574
119894119896) 119906119894(119905minus
119896 119909)
119905 = 119905119896 119896 isin Z
+ 119909 isin Ω
120597119906119894(119905 119909)
120597119899= 0 119905 ge 119905
0 119909 isin 120597Ω
119906119894(1199050+ 119904 119909) = 120595
119894(119904 119909)
minus 120591119895le 119904 le 0 119909 isin Ω
(44)
where 119894 = 1 2 1198881(119905) = 26 119888
2(119905) = 208 119886
11(119905) = minus1 minus cos(119905)
11988612(119905) = 1 + cos(119905) 119886
21(119905) = 1 + sin(119905) 119886
22(119905) = minus1 minus sin(119905)
1198631= 8119863
2= 4 120597119906
119894(119905 119909)120597119899 = 0 (119905 ge 119905
0 119909 = 0 2120587) 120574
1119896= 04
1205742119896
= 02 1205951(sdot) = 120595
1(sdot) = 5 119887
11(119905) = 119887
21(119905) = cos(119905) 119887
12(119905) =
11988722(119905) = minus cos(119905) 119869
1(119905) = 119869
2(119905) = 1 119867
11(119905) = 119867
21(119905) = sin(119905)
11986712(119905) = 119867
22(119905) = minus1 + sin(119905) 119879
11(119905) = 119879
21(119905) = minus sin(119905)
11987912(119905) = 119879
22(119905) = 2 + sin(119905) 120591
1(119905) = 120591
2(119905) = 1 119891
119895(119906119895) =
119906119895(119905 119909) (119895 = 1 2) 119892
119895(119906119895(119905 minus 1 119909)) = 119906
119895(119905 minus 1 119909)119890
minus119906119895(119905minus1119909)
(119895 =
1 2) 12057211(119905) = minus128 120572
21(119905) = 120572
12(119905) = minus1 + cos(119905) 120572
22(119905) =
minus10 12057311(119905) = 128 120573
12(119905) = minus1 + sin(119905) = 120573
21(119905) 12057322(119905) =
8 Abstract and Applied Analysis
6
4
2
0minus5
05
1015
20 76
54
32
10
1199062
119909
119905
5 43
21
119909
Figure 3 State response 1199062(119905 119909) of model (44) without impulsiveeffects
6
4
2
0minus5
05
1015
20 76
54
32
10
1199062
119909
119905
5 43
21
119909
Figure 4 State response 1199062(119905 119909) of model (44) with impulsiveeffects
10 119907119895(119905) = sin(119905) We assume that there exists 119902 = 6 such
that 119905119896+ 2120587 = 119905
119896+119902 Obviously 119891
1 1198912 1198921 and 119892
2satisfy the
assumption (H1) with 1198651= 1198652= 1198661= 1198662= 1 and (H2) and
(H3) are satisfied with a common positive period 2120587
119862 minus (119860 + 120572 + 120573)119865 =[[
[
2
50
04
5
]]
]
(45)
is a nonsingular 119872-matrix The conditions of Theorem 6 aresatisfied hence there exists exactly one 2120596-periodic solutionof the model and all other solutions of the model convergeexponentially to it as 119905 rarr +infin Furthermore the exponentialconverging index can be calculated as120582 = 0021 because here120578119896= 1 and 120578 = 0 The simulation results are shown in Figures
1 2 3 and 4 respectively
5 Conclusions
In this paper periodicity and global exponential stability ofa class of FIRDDCNN model with variable both coefficients
and delays have been investigated By using Halanayrsquos delaydifferential inequality 119872-matrix theory and analytic meth-ods some new sufficient conditions have been established toguarantee the existence uniqueness and global exponentialstability of the periodic solution Moreover the exponentialconvergence rate index can be estimated An example andits simulation have been given to show the effectiveness ofthe obtained results In particular the differentiability ofthe time-varying delays has been removed The dynamicbehaviors of fuzzy neural networks with the property ofexponential periodicity are of great importance inmany areassuch as learning systems
Acknowledgments
The authors would like to thank the editors and the anony-mous reviewers for their valuable comments and construc-tive suggestions This research is supported by the NaturalScience Foundation of Guangxi Autonomous Region (no2012GXNSFBA053003) the National Natural Science Foun-dations of China (60973048 61272077 60974025 60673101and 60939003) National 863 Plan Project (2008 AA04Z4012009 AA043404) the Natural Science Foundation of Shan-dong Province (no Y2007G30) the Scientific andTechnolog-ical Project of Shandong Province (no 2007GG3WZ04016)the Science Foundation of Harbin Institute of Technology(Weihai) (HIT (WH) 200807) theNatural ScientificResearchInnovation Foundation in Harbin Institute of Technology(HIT NSRIF 2001120) the China Postdoctoral ScienceFoundation (2010048 1000) and the Shandong ProvincialKey Laboratory of Industrial Control Technique (QingdaoUniversity)
References
[1] LOChua andL Yang ldquoCellular neural networks theoryrdquo IEEETransactions on Circuits and Systems vol 35 no 10 pp 1257ndash1272 1988
[2] L O Chua and L Yang ldquoCellular neural networks applica-tionsrdquo IEEE Transactions on Circuits and Systems vol 35 no10 pp 1273ndash1290 1988
[3] T Roska and L O Chua ldquoCellular neural networks withnon-linear and delay-type template elements and non-uniformgridsrdquo International Journal of Circuit Theory and Applicationsvol 20 no 5 pp 469ndash481 1992
[4] T Yang L B Yang C W Wu and L O Chua ldquoFuzzycellular neural networks theoryrdquo in Proceedings of the 4th IEEEInternational Workshop on Cellular Neural Networks and TheirApplications (CNNA rsquo96) pp 181ndash186 June 1996
[5] T Yang and L-B Yang ldquoThe global stability of fuzzy cellularneural networkrdquo IEEE Transactions on Circuits and Systems Ivol 43 no 10 pp 880ndash883 1996
[6] T Yang L B Yang C W Wu and L O Chua ldquoFuzzy cellularneural networks applicationsrdquo in Proceedings of the 4th IEEEInternational Workshop on Cellular Neural Networks and TheirApplications (CNNA rsquo96) pp 225ndash230 June 1996
[7] S T Wang K F L Chung and F Duan ldquoApplying theimproved fuzzy cellular neural network IFCNN to white bloodcell detectionrdquo Neurocomputing vol 70 no 7ndash9 pp 1348ndash13592007
Abstract and Applied Analysis 9
[8] T G Barbounis and J B Theocharis ldquoA locally recurrent fuzzyneural network with application to the wind speed predictionusing spatial correlationrdquo Neurocomputing vol 70 no 7ndash9 pp1525ndash1542 2007
[9] H Zhao ldquoGlobal exponential stability and periodicity of cellularneural networks with variable delaysrdquoPhysics Letters A vol 336no 4-5 pp 331ndash341 2005
[10] S Townley A Ilchmann M G Weiss et al ldquoExistence andlearning of oscillations in recurrent neural networksrdquo IEEETransactions on Neural Networks vol 11 no 1 pp 205ndash2142000
[11] Z Huang and Y Xia ldquoExponential periodic attractor ofimpulsive BAM networks with finite distributed delaysrdquo ChaosSolitons amp Fractals vol 39 no 1 pp 373ndash384 2009
[12] Y Liu and W Tang ldquoExponential stability of fuzzy cellularneural networkswith constant and time-varying delaysrdquoPhysicsLetters A vol 323 no 3-4 pp 224ndash233 2004
[13] K Yuan J Cao and J Deng ldquoExponential stability and periodicsolutions of fuzzy cellular neural networks with time-varyingdelaysrdquoNeurocomputing vol 69 no 13ndash15 pp 1619ndash1627 2006
[14] T Huang ldquoExponential stability of fuzzy cellular neural net-works with distributed delayrdquo Physics Letters A vol 351 no 1-2pp 48ndash52 2006
[15] G Bao S Wen and Z Zeng ldquoRobust stability analysis of inter-val fuzzy Cohen-Grossberg neural networks with piecewiseconstant argument of generalized typerdquo Neural Networks vol33 pp 32ndash41 2012
[16] H Zhao N Ding and L Chen ldquoAlmost sure exponential sta-bility of stochastic fuzzy cellular neural networks with delaysrdquoChaos Solitons amp Fractals vol 40 no 4 pp 1653ndash1659 2009
[17] W Ding M Han and M Li ldquoExponential lag synchronizationof delayed fuzzy cellular neural networkswith impulsesrdquoPhysicsLetters A vol 373 no 8-9 pp 832ndash837 2009
[18] L Chen and H Zhao ldquoStability analysis of stochastic fuzzycellular neural networks with delaysrdquo Neurocomputing vol 72no 1ndash3 pp 436ndash444 2008
[19] T Huang ldquoExponential stability of delayed fuzzy cellular neuralnetworks with diffusionrdquo Chaos Solitons amp Fractals vol 31 no3 pp 658ndash664 2007
[20] J Wang and J G Lu ldquoGlobal exponential stability of fuzzycellular neural networks with delays and reaction-diffusiontermsrdquo Chaos Solitons amp Fractals vol 38 no 3 pp 878ndash8852008
[21] Q Song and Z Wang ldquoDynamical behaviors of fuzzy reaction-diffusion periodic cellular neural networks with variable coeffi-cients and delaysrdquo Applied Mathematical Modelling vol 33 no9 pp 3533ndash3545 2009
[22] C Wang Y kao and G Yang ldquoExponential stability ofimpulsive stochastic fuzzy reaction-diffusion Cohen-Grossbergneural networks with mixed delaysrdquo Neurocomputing vol 89pp 55ndash63 2012
[23] T Serrano-Gotarredona and B Linares-Barranco ldquoLog-Domain Implementation of Complex Dynamics Reaction-Diffusion Neural Networksrdquo IEEE Transactions on NeuralNetworks vol 14 no 5 pp 1337ndash1355 2003
[24] J Liang and J Cao ldquoGlobal exponential stability of reaction-diffusion recurrent neural networks with time-varying delaysrdquoPhysics Letters A vol 314 no 5-6 pp 434ndash442 2003
[25] L Wang and D Xu ldquoAsymptotic behavior of a class of reaction-diffusion equations with delaysrdquo Journal of Mathematical Anal-ysis and Applications vol 281 no 2 pp 439ndash453 2003
[26] L Wang and D Xu ldquoGlobal exponential stability of Hopfieldreaction-diffusion neural networks with time-varying delaysrdquoScience in China Series F vol 46 no 6 pp 466ndash474 2003
[27] Z Yang and D Xu ldquoGlobal dynamics for non-autonomousreaction-diffusion neural networks with time-varying delaysrdquoTheoretical Computer Science vol 403 no 1 pp 3ndash10 2008
[28] Y G Kao C C Gao and D K Wang ldquoGlobal exponentialstability of reaction-diffusion Hopfield neural networks withcontinuously distributed delaysrdquo Mathematica Applicata vol21 no 3 pp 457ndash462 2008 (Chinese)
[29] Y-G Kao J-F Guo C-H Wang and X-Q Sun ldquoDelay-dependent robust exponential stability of Markovian jump-ing reaction-diffusion Cohen-Grossberg neural networks withmixed delaysrdquo Journal of the Franklin Institute vol 349 no 6pp 1972ndash1988 2012
[30] H Zhao and Z Mao ldquoBoundedness and stability of nonau-tonomous cellular neural networks with reaction-diffusiontermsrdquo Mathematics and Computers in Simulation vol 79 no5 pp 1603ndash1617 2009
[31] H Zhao and G Wang ldquoExistence of periodic oscillatorysolution of reaction-diffusion neural networks with delaysrdquoPhysics Letters A vol 343 no 5 pp 372ndash383 2005
[32] Q Gan ldquoExponential synchronization of stochastic Cohen-Grossberg neural networks withmixed time-varying delays andreaction-diffusion via periodically intermittent controlrdquoNeuralNetworks vol 31 pp 12ndash21 2012
[33] V Lakshmikantham D D Baınov and P S Simeonov Theoryof Impulsive Differential Equations vol 6 of Series in ModernApplied Mathematics World Scientific Teaneck NJ USA 1989
[34] D D Bainov and P S Simeonov Impulsive Differential Equa-tions Periodic Solutions and Applications Longman HarlowUK 1993
[35] X Song X Xin andWHuang ldquoExponential stability of delayedand impulsive cellular neural networks with partially Lipschitzcontinuous activation functionsrdquo Neural Networks vol 29-30pp 80ndash90 2012
[36] Z Yang and D Xu ldquoExistence and exponential stability ofperiodic solution for impulsive delay differential equations andapplicationsrdquo Nonlinear Analysis Theory Methods amp Applica-tions vol 64 no 1 pp 130ndash145 2006
[37] X Yang X Liao D J Evans and Y Tang ldquoExistence andstability of periodic solution in impulsive Hopfield neuralnetworks with finite distributed delaysrdquo Physics Letters A vol343 no 1ndash3 pp 108ndash116 2005
[38] Y Li and L Lu ldquoGlobal exponential stability and existenceof periodic solution of Hopfield-type neural networks withimpulsesrdquo Physics Letters A vol 333 no 1-2 pp 62ndash71 2004
[39] Y Li L Zhu andP Liu ldquoExistence and stability of periodic solu-tions of delayed cellular neural networksrdquo Nonlinear AnalysisReal World Applications vol 7 no 2 pp 225ndash234 2006
[40] Z H Guan and G Chen ldquoOn delayed impulsive Hopfieldneural networksrdquo Neural Networks vol 12 no 2 pp 273ndash2801999
[41] Z H Guan J Lam andG Chen ldquoOn impulsive autoassociativeneural networksrdquo Neural Networks vol 13 no 1 pp 63ndash692000
[42] W Ding ldquoSynchronization of delayed fuzzy cellular neuralnetworks with impulsive effectsrdquo Communications in NonlinearScience and Numerical Simulation vol 14 no 11 pp 3945ndash39522009
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
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OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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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
Abstract and Applied Analysis 5
Choosing 0 lt 120582 lt min1205821 120582
119899 we have
120585119894(120582119894minus 119888119894)
+
119899
sum
119895=1
120585119895(10038161003816100381610038161003816119886119894119895
10038161003816100381610038161003816119865119895+ 119890120591120582119894 (10038161003816100381610038161003816120572119894119895
10038161003816100381610038161003816+10038161003816100381610038161003816120573119894119895
10038161003816100381610038161003816) 119866119895) lt 0
119894 = 1 119899
(14)
That is
120582120585 minus (119862 minus 119860119865) 120585 + (120572 + 120573)119866120585119890minus120582119905
lt 0 (15)
Furthermore choose a positive scalar 119901 large enough suchthat
119901119890minus120582119905
120585 gt (1 1 1)T 119905 isin [minus120591 0] (16)
For any 120576 gt 0 let
119903 (119905) = 119901119890minus120582119905
(1003817100381710038171003817120601 minus 120595
10038171003817100381710038172+ 120576) 120585 119905
0le 119905 lt 119905
1 (17)
From (15)ndash(17) we obtain
119863+
119903 (119905) gt minus (119862 minus 119860119865) 119903 (119905) + (120572 + 120573)119866119903119904
(119905)
= 119881 (119905 119903 (119905) 119903119904
(119905)) 1199050le 119905 lt 119905
1
(18)
where 119903119904(119905) = (119903119904
1(119905) 119903
119904
119899(119905))
T and 119903119904
119894(119905) = sup
minus120591le119904le0119901119890minus120582(119905+119904)
(120601 minus 1205932+ 120576)120585119894 It is easy to verify that 119881(119905 119903(119905) 119903
119904
(119905)) is an119872-function It follows also from (16) and (17) that
100381710038171003817100381711988011989410038171003817100381710038172
le1003817100381710038171003817120601 minus 120593
10038171003817100381710038172lt 119901119890minus120582119905
120585119894
1003817100381710038171003817120601 minus 12059310038171003817100381710038172
lt 119903119894(119905) 119905 isin [minus120591 0] 119894 = 1 2 119899
(19)
Denote
119880
= (10038171003817100381710038171199061 (119905 119909 120601) minus 119906
1(119905 119909 120595)
10038171003817100381710038172
1003817100381710038171003817119906119899 (119905 119909 120601) minus 119906119899(119905 119909 120595)
10038171003817100381710038172)T
119880(119904)
= (10038171003817100381710038171199061 (119905 119909 120601) minus 119906
1(119905 119909 120595)
1003817100381710038171003817
(119904)
2
1003817100381710038171003817119906119899 (119905 119909 120601) minus 119906119899(119905 119909 120595)
1003817100381710038171003817
(119904)
2)
T
(20)
where 119880119894(119904)
2= sup
minus120591le119904le0119906119894(119905 + 119904 119909 120601) minus 119906
119894(119905 + 119904 119909 120595)
2
then
119880
lt 119903 (119905) 119905 isin [minus120591 0] (21)
From (10) we can obtain
119863+
119880
le minus (119862 minus 119860119865)119880
+ (120572 + 120573)119866119880(119904)
= 119881 (119905 119880
119880(119904)
) 119905 = 119905119896
(22)
Now it follows from (18)ndash(22) and Lemma 5 that
119880
lt 119903 (119905) = 119901119890minus120582119905
(1003817100381710038171003817120601 minus 120595
10038171003817100381710038172+ 120576) 120585
1199050le 119905 lt 119905
1
(23)
Letting 120576 rarr 0 we have
119880
le 1199011205851003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890minus120582119905
1199050le 119905 lt 119905
1 (24)
And moreover from (24) we get
(
119899
sum
119894=1
10038171003817100381710038171198801198941003817100381710038171003817
2
2)
12
le 119901(
119899
sum
119894=1
1205852
119894)
12
1003817100381710038171003817120601 minus 12059510038171003817100381710038172119890minus120582119905
1199050le 119905 lt 119905
1
(25)
Let = 119901(sum119899
119894=11205852
119894)12 then ge 1 Define119882(119905) = 119906
119905(119909 120601)minus
119906119905(119909 120595)
2 it follows from (25) and the definitions of 119906
119905(120601 119909)
and 119906119905(120595 119909) that
119882(119905) =1003817100381710038171003817119906119905(119909 120601) minus 119906
119905(119909 120595)
10038171003817100381710038172
le 1003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890minus120582119905
1199050le 119905 lt 119905
1
(26)
It is easily observed that
119882(119905) le 1003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890minus120582119905
minus120591 le 119905 le 1199050= 0 (27)
Because (26) holds we can suppose that for 119897 le 119896 inequality
119882(119905) le 1205780sdot sdot sdot 120578119897minus1
1003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890minus120582119905
119905119897minus1
le 119905 lt 119905119897
(28)
holds where 1205780= 1 When 119897 = 119896 + 1 we note (H5) that
119882(119905119896) =
10038171003817100381710038171003817119906119905119896
(119909 120601) minus 119906119905119896
(119909 120595)100381710038171003817100381710038172
=10038171003817100381710038171003817ℎ119896(119906minus
119905119896
(119909 120601)) minus ℎ119896(119906minus
119905119896
(119909 120595))100381710038171003817100381710038172
le 120588 (Γ2
119896)10038171003817100381710038171003817119906minus
119905119896
(119909 120601) minus 119906minus
119905119896
(119909 120595)100381710038171003817100381710038172
= 120588 (Γ2
119896)119882 (119905
minus
119896)
le 1205780sdot sdot sdot 120578119897minus1
120588 (Γ2
119896)
1003817100381710038171003817120601 minus 12059510038171003817100381710038172119890minus120582119905119896
le 1205780sdot sdot sdot 120578119897minus1
120578119896120588 (Γ2
119896)
1003817100381710038171003817120601 minus 12059510038171003817100381710038172119890minus120582119905119896
(29)
where 120588(Γ2
119896) is the spectral radius of Γ
2
119896 Let 119872 =
max 120588(Γ2
119896) by (28) (29) and 120578 ge 1 we obtain
119882(119905) le 1205780sdot sdot sdot 120578119897minus1
120578119896119872
1003817100381710038171003817120601 minus 12059510038171003817100381710038172119890minus120582119905
119905119896minus 120591 le 119905 le 119905
119896
(30)
Combining (10) (17) (30) and Lemma 5 we get
119882(119905) le 1205780sdot sdot sdot 120578119897minus1
120578119896119872
1003817100381710038171003817120601 minus 12059510038171003817100381710038172119890minus120582119905
119905119896le 119905 lt 119905
119896+1 119896 isin 119873
+
(31)
Applying mathematical induction we conclude that
119882(119905) le 1205780sdot sdot sdot 120578119897minus1
1198721003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890minus120582119905
119905119896minus1
le 119905 lt 119905119896 119896 isin 119873
+
(32)
6 Abstract and Applied Analysis
From (H6) and (32) we have
119882(119905) le 1198901205781199051119890120578(1199052minus1199051)
sdot sdot sdot 119890120578(119905119896minus1minus119905119896minus2)
times1198721003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890minus120582119905
le 1198721003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890120578119905
119890minus120582119905
= 1198721003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890minus(120582minus120578)119905
119905119896minus1
le 119905 lt 119905119896 119896 isin 119873
+
(33)
This means that1003817100381710038171003817119906119905(119909 120601) minus 119906
119905(119909 120595)
10038171003817100381710038172
le 1198721003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890minus(120582minus120578)119905
le 1198721003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890minus(120582minus120578)(119905minus120591)
119905 ge 1199050
(34)
choosing a positive integer119873 such that
119872119890minus(120582minus120578)(119873120596minus120591)
le1
6 (35)
Define a Poincare mappingD Γ rarr Γ by
D (120601) = 119906120596(119909 120601) (36)
Then
D119873
(120601) = 119906119873120596
(119909 120601) (37)
Setting 119905 = 119873120596 in (34) from (35) and (37) we have
10038171003817100381710038171003817D119873
(120601) minusD119873
(120595)100381710038171003817100381710038172
le1
6
1003817100381710038171003817120601 minus 12059510038171003817100381710038172 (38)
which implies thatD119873 is a contraction mapping Thus thereexists a unique fixed point 120601lowast isin Γ such that
D119873
(D (120601lowast
)) = D (D119873
(120601lowast
)) = D (120601lowast
) (39)
From (37) we know that D(120601lowast
) is also a fixed point of D119873and then it follows from the uniqueness of the fixed point that
D (120601lowast
) = 120601lowast
that is 119906120596(119909 120601lowast
) = 120601lowast
(40)
Let 119906(119905 119909 120601lowast
) be a solution of the model (1) then 119906(119905 +
120596 119909 120601lowast
) is also a solution of the model (1) Obviously
119906119905+120596
(119909 120601lowast
) = 119906119905(119906120596(119909 120601lowast
)) = 119906119905(119909 120601lowast
) (41)
for all 119905 ge 1199050 Hence 119906(119905 + 120596 119909 120601
lowast
) = 119906(119905 119909 120601lowast
) whichshows that 119906(119905 119909 120601lowast) is exactly one 120596-periodic solution ofmodel (1) It is easy to see that all other solutions of model (1)converge to this periodic solution exponentially as 119905 rarr +infinand the exponential convergence rate index is 120582minus120578The proofis completed
Remark 7 When 119888119894(119905) = 119888
119894 119886119894119895(119905) = 119886
119894119895 119887119894119895(119905) = 119887
119894119895 120572119894119895(119905) = 120572
119894119895
120573119894119895(119905) = 120573
119894119895 119879119894119895(119905) = 119879
119894119895119867119894119895(119905) = 119867
119894119895 119907119894(119905) = 119907
119894 119868119894(119905) = 119868
119894 and
120591119905= 120591119894(119888119894 119886119894119895 119887119894119895 120572119894119895 120573119894119895 119879119894119895 119867119894119895 119907119894 119868119894 and 120591
119894are constants)
then the model (1) is changed into
120597119906119894(119905 119909)
120597119905=
119898
sum
119897=1
120597
120597119909119897
(119863119894
120597119906119894(119905 119909)
120597119909119897
)
minus 119888119894119906119894(119905 119909) +
119899
sum
119895=1
119886119894119895119891119895(119906119895(119905 119909))
+
119899
sum
119895=1
119887119894119895119907119895+ 119869119894
+
119899
⋀
119895=1
120572119894119895119892119895(119906119895(119905 minus 120591119895(119905) 119909))
+
119899
⋁
119895=1
120573119894119895119892119895(119906119895(119905 minus 120591119895(119905) 119909))
+
119899
⋀
119895=1
119879119894119895119907119895+
119899
⋁
119895=1
119867119894119895119907119895 119905 = 119905
119896 119909 isin Ω
119906119894(119905+
119896 119909) minus 119906
119894(119905minus
119896 119909) = 119868
119894119896(119906119894(119905minus
119896 119909))
119905 = 119905119896 119896 isin Z
+ 119909 isin Ω
120597119906119894(119905 119909)
120597119899= 0 119905 ge 119905
0 119909 isin 120597Ω
119906119894(1199050+ 119904 119909) = 120595
119894(119904 119909) minus120591
119895le 119904 le 0 119909 isin Ω
(42)
For any positive constant 120596 ge 0 we have 119888119894(119905 + 120596) = 119888
119894(119905)
119886119894119895(119905 + 120596) = 119886
119894119895(119905) 119887119894119895(119905 + 120596) = 119887
119894119895(119905) 120572119894119895(119905 + 120596) = 120572
119894119895(119905)
120573119894119895(119905 + 120596) = 120573
119894119895(119905) 119879119894119895(119905 + 120596) = 119879
119894119895(119905) 119867
119894119895(119905 + 120596) = 119867
119894119895(119905)
119907119894(119905+120596) = 119907
119894(119905) 119868119894(119905+120596) = 119868
119894(119905) and 120591
119894(119905+120596) = 120591
119894(119905) for 119905 ge 119905
0
Thus the sufficient conditions inTheorem 6 are satisfied
Remark 8 If 119868119896(sdot) = 0 the model (1) is changed into
120597119906119894(119905 119909)
120597119905=
119898
sum
119897=1
120597
120597119909119897
(119863119894
120597119906119894(119905 119909)
120597119909119897
)
minus 119888119894(119905) 119906119894(119905 119909) +
119899
sum
119895=1
119886119894119895(119905) 119891119895(119906119895(119905 119909))
+
119899
sum
119895=1
119887119894119895(119905) 119907119895(119905) + 119869
119894(119905)
+
119899
⋀
119895=1
120572119894119895(119905) 119892119895(119906119895(119905 minus 120591119895(119905) 119909))
+
119899
⋁
119895=1
120573119894119895(119905) 119892119895(119906119895(119905 minus 120591119895(119905) 119909))
Abstract and Applied Analysis 7
6
4
2
0minus5
05
1015
20 76
54
32
10
1199061
119909
119905
5 43
21
Figure 1 State response 1199061(119905 119909) of model (44) without impulsiveeffects
+
119899
⋀
119895=1
119879119894119895(119905) 119907119895(119905) +
119899
⋁
119895=1
119867119894119895(119905) 119907119895(119905)
119905 = 119905119896 119909 isin Ω
120597119906119894(119905 119909)
120597119899= 0 119905 ge 119905
0 119909 isin 120597Ω
119906119894(1199050+ 119904 119909) = 120595
119894(119904 119909) minus120591
119895le 119904 le 0 119909 isin Ω
(43)
which has been discussed in [22] As Song and Wang havepointed out the model (43) is more general than some well-studied fuzzy neural networks For example when 119888
119894(119905) gt
0 119886119894119895(119905) 119887119894119895(119905) 120572119894119895(119905) 120573119894119895(119905) 119879119894119895(119905)119867119894119895(119905) 119907119894(119905) and 119868
119894(119905) are all
constants the model in (43) reduces the model which hasbeen studied by Huang [19] Moreover if 119863
119894= 0 120591
119894(119905) = 0
119891119894(120579) = 119892
119894(120579) = (12)(|120579 + 1| minus |120579 minus 1|) (119894 = 1 119899) then
model (42) covers the model studied by Yang et al [4 5] as aspecial case If119863
119894= 0 and 120591
119895(119905) is assumed to be differentiable
for 119894 119895 = 1 2 119899 then model (43) can be specialized tothe model investigated in Liu and Tang [12] and Yuan et al[13] Obviously our results are less conservative than that ofthe above-mentioned literature because they do not considerimpulsive effects
4 Numerical Examples
Example 9 Consider a two-neuron FIRDDCNNmodel
120597119906119894(119905 119909)
120597119905=
119898
sum
119897=1
120597
120597119909119897
(119863119894
120597119906119894(119905 119909)
120597119909119897
)
minus 119888119894(119905) 119906119894(119905 119909)
6
4
2
0minus5
05
1015
20 76
54
32
10
1199061
119909
119905
Figure 2 State response 1199061(119905 119909) of model (44) with impulsiveeffects
+
2
sum
119895=1
119886119894119895(119905) 119891119895(119906119895(119905 119909))
+
2
sum
119895=1
119887119894119895(119905) 119907119895(119905) + 119869
119894(119905)
+
2
⋀
119895=1
120572119894119895(119905) 119892119895(119906119895(119905 minus 120591119895(119905) 119909))
+
2
⋁
119895=1
120573119894119895(119905) 119892119895(119906119895(119905 minus 120591119895(119905) 119909))
+
2
⋀
119895=1
119879119894119895(119905) 119907119895(119905) +
2
⋁
119895=1
119867119894119895(119905) 119907119895(119905)
119905 = 119905119896 119909 isin Ω
119906119894(119905+
119896 119909) = (1 minus 120574
119894119896) 119906119894(119905minus
119896 119909)
119905 = 119905119896 119896 isin Z
+ 119909 isin Ω
120597119906119894(119905 119909)
120597119899= 0 119905 ge 119905
0 119909 isin 120597Ω
119906119894(1199050+ 119904 119909) = 120595
119894(119904 119909)
minus 120591119895le 119904 le 0 119909 isin Ω
(44)
where 119894 = 1 2 1198881(119905) = 26 119888
2(119905) = 208 119886
11(119905) = minus1 minus cos(119905)
11988612(119905) = 1 + cos(119905) 119886
21(119905) = 1 + sin(119905) 119886
22(119905) = minus1 minus sin(119905)
1198631= 8119863
2= 4 120597119906
119894(119905 119909)120597119899 = 0 (119905 ge 119905
0 119909 = 0 2120587) 120574
1119896= 04
1205742119896
= 02 1205951(sdot) = 120595
1(sdot) = 5 119887
11(119905) = 119887
21(119905) = cos(119905) 119887
12(119905) =
11988722(119905) = minus cos(119905) 119869
1(119905) = 119869
2(119905) = 1 119867
11(119905) = 119867
21(119905) = sin(119905)
11986712(119905) = 119867
22(119905) = minus1 + sin(119905) 119879
11(119905) = 119879
21(119905) = minus sin(119905)
11987912(119905) = 119879
22(119905) = 2 + sin(119905) 120591
1(119905) = 120591
2(119905) = 1 119891
119895(119906119895) =
119906119895(119905 119909) (119895 = 1 2) 119892
119895(119906119895(119905 minus 1 119909)) = 119906
119895(119905 minus 1 119909)119890
minus119906119895(119905minus1119909)
(119895 =
1 2) 12057211(119905) = minus128 120572
21(119905) = 120572
12(119905) = minus1 + cos(119905) 120572
22(119905) =
minus10 12057311(119905) = 128 120573
12(119905) = minus1 + sin(119905) = 120573
21(119905) 12057322(119905) =
8 Abstract and Applied Analysis
6
4
2
0minus5
05
1015
20 76
54
32
10
1199062
119909
119905
5 43
21
119909
Figure 3 State response 1199062(119905 119909) of model (44) without impulsiveeffects
6
4
2
0minus5
05
1015
20 76
54
32
10
1199062
119909
119905
5 43
21
119909
Figure 4 State response 1199062(119905 119909) of model (44) with impulsiveeffects
10 119907119895(119905) = sin(119905) We assume that there exists 119902 = 6 such
that 119905119896+ 2120587 = 119905
119896+119902 Obviously 119891
1 1198912 1198921 and 119892
2satisfy the
assumption (H1) with 1198651= 1198652= 1198661= 1198662= 1 and (H2) and
(H3) are satisfied with a common positive period 2120587
119862 minus (119860 + 120572 + 120573)119865 =[[
[
2
50
04
5
]]
]
(45)
is a nonsingular 119872-matrix The conditions of Theorem 6 aresatisfied hence there exists exactly one 2120596-periodic solutionof the model and all other solutions of the model convergeexponentially to it as 119905 rarr +infin Furthermore the exponentialconverging index can be calculated as120582 = 0021 because here120578119896= 1 and 120578 = 0 The simulation results are shown in Figures
1 2 3 and 4 respectively
5 Conclusions
In this paper periodicity and global exponential stability ofa class of FIRDDCNN model with variable both coefficients
and delays have been investigated By using Halanayrsquos delaydifferential inequality 119872-matrix theory and analytic meth-ods some new sufficient conditions have been established toguarantee the existence uniqueness and global exponentialstability of the periodic solution Moreover the exponentialconvergence rate index can be estimated An example andits simulation have been given to show the effectiveness ofthe obtained results In particular the differentiability ofthe time-varying delays has been removed The dynamicbehaviors of fuzzy neural networks with the property ofexponential periodicity are of great importance inmany areassuch as learning systems
Acknowledgments
The authors would like to thank the editors and the anony-mous reviewers for their valuable comments and construc-tive suggestions This research is supported by the NaturalScience Foundation of Guangxi Autonomous Region (no2012GXNSFBA053003) the National Natural Science Foun-dations of China (60973048 61272077 60974025 60673101and 60939003) National 863 Plan Project (2008 AA04Z4012009 AA043404) the Natural Science Foundation of Shan-dong Province (no Y2007G30) the Scientific andTechnolog-ical Project of Shandong Province (no 2007GG3WZ04016)the Science Foundation of Harbin Institute of Technology(Weihai) (HIT (WH) 200807) theNatural ScientificResearchInnovation Foundation in Harbin Institute of Technology(HIT NSRIF 2001120) the China Postdoctoral ScienceFoundation (2010048 1000) and the Shandong ProvincialKey Laboratory of Industrial Control Technique (QingdaoUniversity)
References
[1] LOChua andL Yang ldquoCellular neural networks theoryrdquo IEEETransactions on Circuits and Systems vol 35 no 10 pp 1257ndash1272 1988
[2] L O Chua and L Yang ldquoCellular neural networks applica-tionsrdquo IEEE Transactions on Circuits and Systems vol 35 no10 pp 1273ndash1290 1988
[3] T Roska and L O Chua ldquoCellular neural networks withnon-linear and delay-type template elements and non-uniformgridsrdquo International Journal of Circuit Theory and Applicationsvol 20 no 5 pp 469ndash481 1992
[4] T Yang L B Yang C W Wu and L O Chua ldquoFuzzycellular neural networks theoryrdquo in Proceedings of the 4th IEEEInternational Workshop on Cellular Neural Networks and TheirApplications (CNNA rsquo96) pp 181ndash186 June 1996
[5] T Yang and L-B Yang ldquoThe global stability of fuzzy cellularneural networkrdquo IEEE Transactions on Circuits and Systems Ivol 43 no 10 pp 880ndash883 1996
[6] T Yang L B Yang C W Wu and L O Chua ldquoFuzzy cellularneural networks applicationsrdquo in Proceedings of the 4th IEEEInternational Workshop on Cellular Neural Networks and TheirApplications (CNNA rsquo96) pp 225ndash230 June 1996
[7] S T Wang K F L Chung and F Duan ldquoApplying theimproved fuzzy cellular neural network IFCNN to white bloodcell detectionrdquo Neurocomputing vol 70 no 7ndash9 pp 1348ndash13592007
Abstract and Applied Analysis 9
[8] T G Barbounis and J B Theocharis ldquoA locally recurrent fuzzyneural network with application to the wind speed predictionusing spatial correlationrdquo Neurocomputing vol 70 no 7ndash9 pp1525ndash1542 2007
[9] H Zhao ldquoGlobal exponential stability and periodicity of cellularneural networks with variable delaysrdquoPhysics Letters A vol 336no 4-5 pp 331ndash341 2005
[10] S Townley A Ilchmann M G Weiss et al ldquoExistence andlearning of oscillations in recurrent neural networksrdquo IEEETransactions on Neural Networks vol 11 no 1 pp 205ndash2142000
[11] Z Huang and Y Xia ldquoExponential periodic attractor ofimpulsive BAM networks with finite distributed delaysrdquo ChaosSolitons amp Fractals vol 39 no 1 pp 373ndash384 2009
[12] Y Liu and W Tang ldquoExponential stability of fuzzy cellularneural networkswith constant and time-varying delaysrdquoPhysicsLetters A vol 323 no 3-4 pp 224ndash233 2004
[13] K Yuan J Cao and J Deng ldquoExponential stability and periodicsolutions of fuzzy cellular neural networks with time-varyingdelaysrdquoNeurocomputing vol 69 no 13ndash15 pp 1619ndash1627 2006
[14] T Huang ldquoExponential stability of fuzzy cellular neural net-works with distributed delayrdquo Physics Letters A vol 351 no 1-2pp 48ndash52 2006
[15] G Bao S Wen and Z Zeng ldquoRobust stability analysis of inter-val fuzzy Cohen-Grossberg neural networks with piecewiseconstant argument of generalized typerdquo Neural Networks vol33 pp 32ndash41 2012
[16] H Zhao N Ding and L Chen ldquoAlmost sure exponential sta-bility of stochastic fuzzy cellular neural networks with delaysrdquoChaos Solitons amp Fractals vol 40 no 4 pp 1653ndash1659 2009
[17] W Ding M Han and M Li ldquoExponential lag synchronizationof delayed fuzzy cellular neural networkswith impulsesrdquoPhysicsLetters A vol 373 no 8-9 pp 832ndash837 2009
[18] L Chen and H Zhao ldquoStability analysis of stochastic fuzzycellular neural networks with delaysrdquo Neurocomputing vol 72no 1ndash3 pp 436ndash444 2008
[19] T Huang ldquoExponential stability of delayed fuzzy cellular neuralnetworks with diffusionrdquo Chaos Solitons amp Fractals vol 31 no3 pp 658ndash664 2007
[20] J Wang and J G Lu ldquoGlobal exponential stability of fuzzycellular neural networks with delays and reaction-diffusiontermsrdquo Chaos Solitons amp Fractals vol 38 no 3 pp 878ndash8852008
[21] Q Song and Z Wang ldquoDynamical behaviors of fuzzy reaction-diffusion periodic cellular neural networks with variable coeffi-cients and delaysrdquo Applied Mathematical Modelling vol 33 no9 pp 3533ndash3545 2009
[22] C Wang Y kao and G Yang ldquoExponential stability ofimpulsive stochastic fuzzy reaction-diffusion Cohen-Grossbergneural networks with mixed delaysrdquo Neurocomputing vol 89pp 55ndash63 2012
[23] T Serrano-Gotarredona and B Linares-Barranco ldquoLog-Domain Implementation of Complex Dynamics Reaction-Diffusion Neural Networksrdquo IEEE Transactions on NeuralNetworks vol 14 no 5 pp 1337ndash1355 2003
[24] J Liang and J Cao ldquoGlobal exponential stability of reaction-diffusion recurrent neural networks with time-varying delaysrdquoPhysics Letters A vol 314 no 5-6 pp 434ndash442 2003
[25] L Wang and D Xu ldquoAsymptotic behavior of a class of reaction-diffusion equations with delaysrdquo Journal of Mathematical Anal-ysis and Applications vol 281 no 2 pp 439ndash453 2003
[26] L Wang and D Xu ldquoGlobal exponential stability of Hopfieldreaction-diffusion neural networks with time-varying delaysrdquoScience in China Series F vol 46 no 6 pp 466ndash474 2003
[27] Z Yang and D Xu ldquoGlobal dynamics for non-autonomousreaction-diffusion neural networks with time-varying delaysrdquoTheoretical Computer Science vol 403 no 1 pp 3ndash10 2008
[28] Y G Kao C C Gao and D K Wang ldquoGlobal exponentialstability of reaction-diffusion Hopfield neural networks withcontinuously distributed delaysrdquo Mathematica Applicata vol21 no 3 pp 457ndash462 2008 (Chinese)
[29] Y-G Kao J-F Guo C-H Wang and X-Q Sun ldquoDelay-dependent robust exponential stability of Markovian jump-ing reaction-diffusion Cohen-Grossberg neural networks withmixed delaysrdquo Journal of the Franklin Institute vol 349 no 6pp 1972ndash1988 2012
[30] H Zhao and Z Mao ldquoBoundedness and stability of nonau-tonomous cellular neural networks with reaction-diffusiontermsrdquo Mathematics and Computers in Simulation vol 79 no5 pp 1603ndash1617 2009
[31] H Zhao and G Wang ldquoExistence of periodic oscillatorysolution of reaction-diffusion neural networks with delaysrdquoPhysics Letters A vol 343 no 5 pp 372ndash383 2005
[32] Q Gan ldquoExponential synchronization of stochastic Cohen-Grossberg neural networks withmixed time-varying delays andreaction-diffusion via periodically intermittent controlrdquoNeuralNetworks vol 31 pp 12ndash21 2012
[33] V Lakshmikantham D D Baınov and P S Simeonov Theoryof Impulsive Differential Equations vol 6 of Series in ModernApplied Mathematics World Scientific Teaneck NJ USA 1989
[34] D D Bainov and P S Simeonov Impulsive Differential Equa-tions Periodic Solutions and Applications Longman HarlowUK 1993
[35] X Song X Xin andWHuang ldquoExponential stability of delayedand impulsive cellular neural networks with partially Lipschitzcontinuous activation functionsrdquo Neural Networks vol 29-30pp 80ndash90 2012
[36] Z Yang and D Xu ldquoExistence and exponential stability ofperiodic solution for impulsive delay differential equations andapplicationsrdquo Nonlinear Analysis Theory Methods amp Applica-tions vol 64 no 1 pp 130ndash145 2006
[37] X Yang X Liao D J Evans and Y Tang ldquoExistence andstability of periodic solution in impulsive Hopfield neuralnetworks with finite distributed delaysrdquo Physics Letters A vol343 no 1ndash3 pp 108ndash116 2005
[38] Y Li and L Lu ldquoGlobal exponential stability and existenceof periodic solution of Hopfield-type neural networks withimpulsesrdquo Physics Letters A vol 333 no 1-2 pp 62ndash71 2004
[39] Y Li L Zhu andP Liu ldquoExistence and stability of periodic solu-tions of delayed cellular neural networksrdquo Nonlinear AnalysisReal World Applications vol 7 no 2 pp 225ndash234 2006
[40] Z H Guan and G Chen ldquoOn delayed impulsive Hopfieldneural networksrdquo Neural Networks vol 12 no 2 pp 273ndash2801999
[41] Z H Guan J Lam andG Chen ldquoOn impulsive autoassociativeneural networksrdquo Neural Networks vol 13 no 1 pp 63ndash692000
[42] W Ding ldquoSynchronization of delayed fuzzy cellular neuralnetworks with impulsive effectsrdquo Communications in NonlinearScience and Numerical Simulation vol 14 no 11 pp 3945ndash39522009
Submit your manuscripts athttpwwwhindawicom
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
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Stochastic AnalysisInternational Journal of
6 Abstract and Applied Analysis
From (H6) and (32) we have
119882(119905) le 1198901205781199051119890120578(1199052minus1199051)
sdot sdot sdot 119890120578(119905119896minus1minus119905119896minus2)
times1198721003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890minus120582119905
le 1198721003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890120578119905
119890minus120582119905
= 1198721003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890minus(120582minus120578)119905
119905119896minus1
le 119905 lt 119905119896 119896 isin 119873
+
(33)
This means that1003817100381710038171003817119906119905(119909 120601) minus 119906
119905(119909 120595)
10038171003817100381710038172
le 1198721003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890minus(120582minus120578)119905
le 1198721003817100381710038171003817120601 minus 120595
10038171003817100381710038172119890minus(120582minus120578)(119905minus120591)
119905 ge 1199050
(34)
choosing a positive integer119873 such that
119872119890minus(120582minus120578)(119873120596minus120591)
le1
6 (35)
Define a Poincare mappingD Γ rarr Γ by
D (120601) = 119906120596(119909 120601) (36)
Then
D119873
(120601) = 119906119873120596
(119909 120601) (37)
Setting 119905 = 119873120596 in (34) from (35) and (37) we have
10038171003817100381710038171003817D119873
(120601) minusD119873
(120595)100381710038171003817100381710038172
le1
6
1003817100381710038171003817120601 minus 12059510038171003817100381710038172 (38)
which implies thatD119873 is a contraction mapping Thus thereexists a unique fixed point 120601lowast isin Γ such that
D119873
(D (120601lowast
)) = D (D119873
(120601lowast
)) = D (120601lowast
) (39)
From (37) we know that D(120601lowast
) is also a fixed point of D119873and then it follows from the uniqueness of the fixed point that
D (120601lowast
) = 120601lowast
that is 119906120596(119909 120601lowast
) = 120601lowast
(40)
Let 119906(119905 119909 120601lowast
) be a solution of the model (1) then 119906(119905 +
120596 119909 120601lowast
) is also a solution of the model (1) Obviously
119906119905+120596
(119909 120601lowast
) = 119906119905(119906120596(119909 120601lowast
)) = 119906119905(119909 120601lowast
) (41)
for all 119905 ge 1199050 Hence 119906(119905 + 120596 119909 120601
lowast
) = 119906(119905 119909 120601lowast
) whichshows that 119906(119905 119909 120601lowast) is exactly one 120596-periodic solution ofmodel (1) It is easy to see that all other solutions of model (1)converge to this periodic solution exponentially as 119905 rarr +infinand the exponential convergence rate index is 120582minus120578The proofis completed
Remark 7 When 119888119894(119905) = 119888
119894 119886119894119895(119905) = 119886
119894119895 119887119894119895(119905) = 119887
119894119895 120572119894119895(119905) = 120572
119894119895
120573119894119895(119905) = 120573
119894119895 119879119894119895(119905) = 119879
119894119895119867119894119895(119905) = 119867
119894119895 119907119894(119905) = 119907
119894 119868119894(119905) = 119868
119894 and
120591119905= 120591119894(119888119894 119886119894119895 119887119894119895 120572119894119895 120573119894119895 119879119894119895 119867119894119895 119907119894 119868119894 and 120591
119894are constants)
then the model (1) is changed into
120597119906119894(119905 119909)
120597119905=
119898
sum
119897=1
120597
120597119909119897
(119863119894
120597119906119894(119905 119909)
120597119909119897
)
minus 119888119894119906119894(119905 119909) +
119899
sum
119895=1
119886119894119895119891119895(119906119895(119905 119909))
+
119899
sum
119895=1
119887119894119895119907119895+ 119869119894
+
119899
⋀
119895=1
120572119894119895119892119895(119906119895(119905 minus 120591119895(119905) 119909))
+
119899
⋁
119895=1
120573119894119895119892119895(119906119895(119905 minus 120591119895(119905) 119909))
+
119899
⋀
119895=1
119879119894119895119907119895+
119899
⋁
119895=1
119867119894119895119907119895 119905 = 119905
119896 119909 isin Ω
119906119894(119905+
119896 119909) minus 119906
119894(119905minus
119896 119909) = 119868
119894119896(119906119894(119905minus
119896 119909))
119905 = 119905119896 119896 isin Z
+ 119909 isin Ω
120597119906119894(119905 119909)
120597119899= 0 119905 ge 119905
0 119909 isin 120597Ω
119906119894(1199050+ 119904 119909) = 120595
119894(119904 119909) minus120591
119895le 119904 le 0 119909 isin Ω
(42)
For any positive constant 120596 ge 0 we have 119888119894(119905 + 120596) = 119888
119894(119905)
119886119894119895(119905 + 120596) = 119886
119894119895(119905) 119887119894119895(119905 + 120596) = 119887
119894119895(119905) 120572119894119895(119905 + 120596) = 120572
119894119895(119905)
120573119894119895(119905 + 120596) = 120573
119894119895(119905) 119879119894119895(119905 + 120596) = 119879
119894119895(119905) 119867
119894119895(119905 + 120596) = 119867
119894119895(119905)
119907119894(119905+120596) = 119907
119894(119905) 119868119894(119905+120596) = 119868
119894(119905) and 120591
119894(119905+120596) = 120591
119894(119905) for 119905 ge 119905
0
Thus the sufficient conditions inTheorem 6 are satisfied
Remark 8 If 119868119896(sdot) = 0 the model (1) is changed into
120597119906119894(119905 119909)
120597119905=
119898
sum
119897=1
120597
120597119909119897
(119863119894
120597119906119894(119905 119909)
120597119909119897
)
minus 119888119894(119905) 119906119894(119905 119909) +
119899
sum
119895=1
119886119894119895(119905) 119891119895(119906119895(119905 119909))
+
119899
sum
119895=1
119887119894119895(119905) 119907119895(119905) + 119869
119894(119905)
+
119899
⋀
119895=1
120572119894119895(119905) 119892119895(119906119895(119905 minus 120591119895(119905) 119909))
+
119899
⋁
119895=1
120573119894119895(119905) 119892119895(119906119895(119905 minus 120591119895(119905) 119909))
Abstract and Applied Analysis 7
6
4
2
0minus5
05
1015
20 76
54
32
10
1199061
119909
119905
5 43
21
Figure 1 State response 1199061(119905 119909) of model (44) without impulsiveeffects
+
119899
⋀
119895=1
119879119894119895(119905) 119907119895(119905) +
119899
⋁
119895=1
119867119894119895(119905) 119907119895(119905)
119905 = 119905119896 119909 isin Ω
120597119906119894(119905 119909)
120597119899= 0 119905 ge 119905
0 119909 isin 120597Ω
119906119894(1199050+ 119904 119909) = 120595
119894(119904 119909) minus120591
119895le 119904 le 0 119909 isin Ω
(43)
which has been discussed in [22] As Song and Wang havepointed out the model (43) is more general than some well-studied fuzzy neural networks For example when 119888
119894(119905) gt
0 119886119894119895(119905) 119887119894119895(119905) 120572119894119895(119905) 120573119894119895(119905) 119879119894119895(119905)119867119894119895(119905) 119907119894(119905) and 119868
119894(119905) are all
constants the model in (43) reduces the model which hasbeen studied by Huang [19] Moreover if 119863
119894= 0 120591
119894(119905) = 0
119891119894(120579) = 119892
119894(120579) = (12)(|120579 + 1| minus |120579 minus 1|) (119894 = 1 119899) then
model (42) covers the model studied by Yang et al [4 5] as aspecial case If119863
119894= 0 and 120591
119895(119905) is assumed to be differentiable
for 119894 119895 = 1 2 119899 then model (43) can be specialized tothe model investigated in Liu and Tang [12] and Yuan et al[13] Obviously our results are less conservative than that ofthe above-mentioned literature because they do not considerimpulsive effects
4 Numerical Examples
Example 9 Consider a two-neuron FIRDDCNNmodel
120597119906119894(119905 119909)
120597119905=
119898
sum
119897=1
120597
120597119909119897
(119863119894
120597119906119894(119905 119909)
120597119909119897
)
minus 119888119894(119905) 119906119894(119905 119909)
6
4
2
0minus5
05
1015
20 76
54
32
10
1199061
119909
119905
Figure 2 State response 1199061(119905 119909) of model (44) with impulsiveeffects
+
2
sum
119895=1
119886119894119895(119905) 119891119895(119906119895(119905 119909))
+
2
sum
119895=1
119887119894119895(119905) 119907119895(119905) + 119869
119894(119905)
+
2
⋀
119895=1
120572119894119895(119905) 119892119895(119906119895(119905 minus 120591119895(119905) 119909))
+
2
⋁
119895=1
120573119894119895(119905) 119892119895(119906119895(119905 minus 120591119895(119905) 119909))
+
2
⋀
119895=1
119879119894119895(119905) 119907119895(119905) +
2
⋁
119895=1
119867119894119895(119905) 119907119895(119905)
119905 = 119905119896 119909 isin Ω
119906119894(119905+
119896 119909) = (1 minus 120574
119894119896) 119906119894(119905minus
119896 119909)
119905 = 119905119896 119896 isin Z
+ 119909 isin Ω
120597119906119894(119905 119909)
120597119899= 0 119905 ge 119905
0 119909 isin 120597Ω
119906119894(1199050+ 119904 119909) = 120595
119894(119904 119909)
minus 120591119895le 119904 le 0 119909 isin Ω
(44)
where 119894 = 1 2 1198881(119905) = 26 119888
2(119905) = 208 119886
11(119905) = minus1 minus cos(119905)
11988612(119905) = 1 + cos(119905) 119886
21(119905) = 1 + sin(119905) 119886
22(119905) = minus1 minus sin(119905)
1198631= 8119863
2= 4 120597119906
119894(119905 119909)120597119899 = 0 (119905 ge 119905
0 119909 = 0 2120587) 120574
1119896= 04
1205742119896
= 02 1205951(sdot) = 120595
1(sdot) = 5 119887
11(119905) = 119887
21(119905) = cos(119905) 119887
12(119905) =
11988722(119905) = minus cos(119905) 119869
1(119905) = 119869
2(119905) = 1 119867
11(119905) = 119867
21(119905) = sin(119905)
11986712(119905) = 119867
22(119905) = minus1 + sin(119905) 119879
11(119905) = 119879
21(119905) = minus sin(119905)
11987912(119905) = 119879
22(119905) = 2 + sin(119905) 120591
1(119905) = 120591
2(119905) = 1 119891
119895(119906119895) =
119906119895(119905 119909) (119895 = 1 2) 119892
119895(119906119895(119905 minus 1 119909)) = 119906
119895(119905 minus 1 119909)119890
minus119906119895(119905minus1119909)
(119895 =
1 2) 12057211(119905) = minus128 120572
21(119905) = 120572
12(119905) = minus1 + cos(119905) 120572
22(119905) =
minus10 12057311(119905) = 128 120573
12(119905) = minus1 + sin(119905) = 120573
21(119905) 12057322(119905) =
8 Abstract and Applied Analysis
6
4
2
0minus5
05
1015
20 76
54
32
10
1199062
119909
119905
5 43
21
119909
Figure 3 State response 1199062(119905 119909) of model (44) without impulsiveeffects
6
4
2
0minus5
05
1015
20 76
54
32
10
1199062
119909
119905
5 43
21
119909
Figure 4 State response 1199062(119905 119909) of model (44) with impulsiveeffects
10 119907119895(119905) = sin(119905) We assume that there exists 119902 = 6 such
that 119905119896+ 2120587 = 119905
119896+119902 Obviously 119891
1 1198912 1198921 and 119892
2satisfy the
assumption (H1) with 1198651= 1198652= 1198661= 1198662= 1 and (H2) and
(H3) are satisfied with a common positive period 2120587
119862 minus (119860 + 120572 + 120573)119865 =[[
[
2
50
04
5
]]
]
(45)
is a nonsingular 119872-matrix The conditions of Theorem 6 aresatisfied hence there exists exactly one 2120596-periodic solutionof the model and all other solutions of the model convergeexponentially to it as 119905 rarr +infin Furthermore the exponentialconverging index can be calculated as120582 = 0021 because here120578119896= 1 and 120578 = 0 The simulation results are shown in Figures
1 2 3 and 4 respectively
5 Conclusions
In this paper periodicity and global exponential stability ofa class of FIRDDCNN model with variable both coefficients
and delays have been investigated By using Halanayrsquos delaydifferential inequality 119872-matrix theory and analytic meth-ods some new sufficient conditions have been established toguarantee the existence uniqueness and global exponentialstability of the periodic solution Moreover the exponentialconvergence rate index can be estimated An example andits simulation have been given to show the effectiveness ofthe obtained results In particular the differentiability ofthe time-varying delays has been removed The dynamicbehaviors of fuzzy neural networks with the property ofexponential periodicity are of great importance inmany areassuch as learning systems
Acknowledgments
The authors would like to thank the editors and the anony-mous reviewers for their valuable comments and construc-tive suggestions This research is supported by the NaturalScience Foundation of Guangxi Autonomous Region (no2012GXNSFBA053003) the National Natural Science Foun-dations of China (60973048 61272077 60974025 60673101and 60939003) National 863 Plan Project (2008 AA04Z4012009 AA043404) the Natural Science Foundation of Shan-dong Province (no Y2007G30) the Scientific andTechnolog-ical Project of Shandong Province (no 2007GG3WZ04016)the Science Foundation of Harbin Institute of Technology(Weihai) (HIT (WH) 200807) theNatural ScientificResearchInnovation Foundation in Harbin Institute of Technology(HIT NSRIF 2001120) the China Postdoctoral ScienceFoundation (2010048 1000) and the Shandong ProvincialKey Laboratory of Industrial Control Technique (QingdaoUniversity)
References
[1] LOChua andL Yang ldquoCellular neural networks theoryrdquo IEEETransactions on Circuits and Systems vol 35 no 10 pp 1257ndash1272 1988
[2] L O Chua and L Yang ldquoCellular neural networks applica-tionsrdquo IEEE Transactions on Circuits and Systems vol 35 no10 pp 1273ndash1290 1988
[3] T Roska and L O Chua ldquoCellular neural networks withnon-linear and delay-type template elements and non-uniformgridsrdquo International Journal of Circuit Theory and Applicationsvol 20 no 5 pp 469ndash481 1992
[4] T Yang L B Yang C W Wu and L O Chua ldquoFuzzycellular neural networks theoryrdquo in Proceedings of the 4th IEEEInternational Workshop on Cellular Neural Networks and TheirApplications (CNNA rsquo96) pp 181ndash186 June 1996
[5] T Yang and L-B Yang ldquoThe global stability of fuzzy cellularneural networkrdquo IEEE Transactions on Circuits and Systems Ivol 43 no 10 pp 880ndash883 1996
[6] T Yang L B Yang C W Wu and L O Chua ldquoFuzzy cellularneural networks applicationsrdquo in Proceedings of the 4th IEEEInternational Workshop on Cellular Neural Networks and TheirApplications (CNNA rsquo96) pp 225ndash230 June 1996
[7] S T Wang K F L Chung and F Duan ldquoApplying theimproved fuzzy cellular neural network IFCNN to white bloodcell detectionrdquo Neurocomputing vol 70 no 7ndash9 pp 1348ndash13592007
Abstract and Applied Analysis 9
[8] T G Barbounis and J B Theocharis ldquoA locally recurrent fuzzyneural network with application to the wind speed predictionusing spatial correlationrdquo Neurocomputing vol 70 no 7ndash9 pp1525ndash1542 2007
[9] H Zhao ldquoGlobal exponential stability and periodicity of cellularneural networks with variable delaysrdquoPhysics Letters A vol 336no 4-5 pp 331ndash341 2005
[10] S Townley A Ilchmann M G Weiss et al ldquoExistence andlearning of oscillations in recurrent neural networksrdquo IEEETransactions on Neural Networks vol 11 no 1 pp 205ndash2142000
[11] Z Huang and Y Xia ldquoExponential periodic attractor ofimpulsive BAM networks with finite distributed delaysrdquo ChaosSolitons amp Fractals vol 39 no 1 pp 373ndash384 2009
[12] Y Liu and W Tang ldquoExponential stability of fuzzy cellularneural networkswith constant and time-varying delaysrdquoPhysicsLetters A vol 323 no 3-4 pp 224ndash233 2004
[13] K Yuan J Cao and J Deng ldquoExponential stability and periodicsolutions of fuzzy cellular neural networks with time-varyingdelaysrdquoNeurocomputing vol 69 no 13ndash15 pp 1619ndash1627 2006
[14] T Huang ldquoExponential stability of fuzzy cellular neural net-works with distributed delayrdquo Physics Letters A vol 351 no 1-2pp 48ndash52 2006
[15] G Bao S Wen and Z Zeng ldquoRobust stability analysis of inter-val fuzzy Cohen-Grossberg neural networks with piecewiseconstant argument of generalized typerdquo Neural Networks vol33 pp 32ndash41 2012
[16] H Zhao N Ding and L Chen ldquoAlmost sure exponential sta-bility of stochastic fuzzy cellular neural networks with delaysrdquoChaos Solitons amp Fractals vol 40 no 4 pp 1653ndash1659 2009
[17] W Ding M Han and M Li ldquoExponential lag synchronizationof delayed fuzzy cellular neural networkswith impulsesrdquoPhysicsLetters A vol 373 no 8-9 pp 832ndash837 2009
[18] L Chen and H Zhao ldquoStability analysis of stochastic fuzzycellular neural networks with delaysrdquo Neurocomputing vol 72no 1ndash3 pp 436ndash444 2008
[19] T Huang ldquoExponential stability of delayed fuzzy cellular neuralnetworks with diffusionrdquo Chaos Solitons amp Fractals vol 31 no3 pp 658ndash664 2007
[20] J Wang and J G Lu ldquoGlobal exponential stability of fuzzycellular neural networks with delays and reaction-diffusiontermsrdquo Chaos Solitons amp Fractals vol 38 no 3 pp 878ndash8852008
[21] Q Song and Z Wang ldquoDynamical behaviors of fuzzy reaction-diffusion periodic cellular neural networks with variable coeffi-cients and delaysrdquo Applied Mathematical Modelling vol 33 no9 pp 3533ndash3545 2009
[22] C Wang Y kao and G Yang ldquoExponential stability ofimpulsive stochastic fuzzy reaction-diffusion Cohen-Grossbergneural networks with mixed delaysrdquo Neurocomputing vol 89pp 55ndash63 2012
[23] T Serrano-Gotarredona and B Linares-Barranco ldquoLog-Domain Implementation of Complex Dynamics Reaction-Diffusion Neural Networksrdquo IEEE Transactions on NeuralNetworks vol 14 no 5 pp 1337ndash1355 2003
[24] J Liang and J Cao ldquoGlobal exponential stability of reaction-diffusion recurrent neural networks with time-varying delaysrdquoPhysics Letters A vol 314 no 5-6 pp 434ndash442 2003
[25] L Wang and D Xu ldquoAsymptotic behavior of a class of reaction-diffusion equations with delaysrdquo Journal of Mathematical Anal-ysis and Applications vol 281 no 2 pp 439ndash453 2003
[26] L Wang and D Xu ldquoGlobal exponential stability of Hopfieldreaction-diffusion neural networks with time-varying delaysrdquoScience in China Series F vol 46 no 6 pp 466ndash474 2003
[27] Z Yang and D Xu ldquoGlobal dynamics for non-autonomousreaction-diffusion neural networks with time-varying delaysrdquoTheoretical Computer Science vol 403 no 1 pp 3ndash10 2008
[28] Y G Kao C C Gao and D K Wang ldquoGlobal exponentialstability of reaction-diffusion Hopfield neural networks withcontinuously distributed delaysrdquo Mathematica Applicata vol21 no 3 pp 457ndash462 2008 (Chinese)
[29] Y-G Kao J-F Guo C-H Wang and X-Q Sun ldquoDelay-dependent robust exponential stability of Markovian jump-ing reaction-diffusion Cohen-Grossberg neural networks withmixed delaysrdquo Journal of the Franklin Institute vol 349 no 6pp 1972ndash1988 2012
[30] H Zhao and Z Mao ldquoBoundedness and stability of nonau-tonomous cellular neural networks with reaction-diffusiontermsrdquo Mathematics and Computers in Simulation vol 79 no5 pp 1603ndash1617 2009
[31] H Zhao and G Wang ldquoExistence of periodic oscillatorysolution of reaction-diffusion neural networks with delaysrdquoPhysics Letters A vol 343 no 5 pp 372ndash383 2005
[32] Q Gan ldquoExponential synchronization of stochastic Cohen-Grossberg neural networks withmixed time-varying delays andreaction-diffusion via periodically intermittent controlrdquoNeuralNetworks vol 31 pp 12ndash21 2012
[33] V Lakshmikantham D D Baınov and P S Simeonov Theoryof Impulsive Differential Equations vol 6 of Series in ModernApplied Mathematics World Scientific Teaneck NJ USA 1989
[34] D D Bainov and P S Simeonov Impulsive Differential Equa-tions Periodic Solutions and Applications Longman HarlowUK 1993
[35] X Song X Xin andWHuang ldquoExponential stability of delayedand impulsive cellular neural networks with partially Lipschitzcontinuous activation functionsrdquo Neural Networks vol 29-30pp 80ndash90 2012
[36] Z Yang and D Xu ldquoExistence and exponential stability ofperiodic solution for impulsive delay differential equations andapplicationsrdquo Nonlinear Analysis Theory Methods amp Applica-tions vol 64 no 1 pp 130ndash145 2006
[37] X Yang X Liao D J Evans and Y Tang ldquoExistence andstability of periodic solution in impulsive Hopfield neuralnetworks with finite distributed delaysrdquo Physics Letters A vol343 no 1ndash3 pp 108ndash116 2005
[38] Y Li and L Lu ldquoGlobal exponential stability and existenceof periodic solution of Hopfield-type neural networks withimpulsesrdquo Physics Letters A vol 333 no 1-2 pp 62ndash71 2004
[39] Y Li L Zhu andP Liu ldquoExistence and stability of periodic solu-tions of delayed cellular neural networksrdquo Nonlinear AnalysisReal World Applications vol 7 no 2 pp 225ndash234 2006
[40] Z H Guan and G Chen ldquoOn delayed impulsive Hopfieldneural networksrdquo Neural Networks vol 12 no 2 pp 273ndash2801999
[41] Z H Guan J Lam andG Chen ldquoOn impulsive autoassociativeneural networksrdquo Neural Networks vol 13 no 1 pp 63ndash692000
[42] W Ding ldquoSynchronization of delayed fuzzy cellular neuralnetworks with impulsive effectsrdquo Communications in NonlinearScience and Numerical Simulation vol 14 no 11 pp 3945ndash39522009
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
Abstract and Applied Analysis 7
6
4
2
0minus5
05
1015
20 76
54
32
10
1199061
119909
119905
5 43
21
Figure 1 State response 1199061(119905 119909) of model (44) without impulsiveeffects
+
119899
⋀
119895=1
119879119894119895(119905) 119907119895(119905) +
119899
⋁
119895=1
119867119894119895(119905) 119907119895(119905)
119905 = 119905119896 119909 isin Ω
120597119906119894(119905 119909)
120597119899= 0 119905 ge 119905
0 119909 isin 120597Ω
119906119894(1199050+ 119904 119909) = 120595
119894(119904 119909) minus120591
119895le 119904 le 0 119909 isin Ω
(43)
which has been discussed in [22] As Song and Wang havepointed out the model (43) is more general than some well-studied fuzzy neural networks For example when 119888
119894(119905) gt
0 119886119894119895(119905) 119887119894119895(119905) 120572119894119895(119905) 120573119894119895(119905) 119879119894119895(119905)119867119894119895(119905) 119907119894(119905) and 119868
119894(119905) are all
constants the model in (43) reduces the model which hasbeen studied by Huang [19] Moreover if 119863
119894= 0 120591
119894(119905) = 0
119891119894(120579) = 119892
119894(120579) = (12)(|120579 + 1| minus |120579 minus 1|) (119894 = 1 119899) then
model (42) covers the model studied by Yang et al [4 5] as aspecial case If119863
119894= 0 and 120591
119895(119905) is assumed to be differentiable
for 119894 119895 = 1 2 119899 then model (43) can be specialized tothe model investigated in Liu and Tang [12] and Yuan et al[13] Obviously our results are less conservative than that ofthe above-mentioned literature because they do not considerimpulsive effects
4 Numerical Examples
Example 9 Consider a two-neuron FIRDDCNNmodel
120597119906119894(119905 119909)
120597119905=
119898
sum
119897=1
120597
120597119909119897
(119863119894
120597119906119894(119905 119909)
120597119909119897
)
minus 119888119894(119905) 119906119894(119905 119909)
6
4
2
0minus5
05
1015
20 76
54
32
10
1199061
119909
119905
Figure 2 State response 1199061(119905 119909) of model (44) with impulsiveeffects
+
2
sum
119895=1
119886119894119895(119905) 119891119895(119906119895(119905 119909))
+
2
sum
119895=1
119887119894119895(119905) 119907119895(119905) + 119869
119894(119905)
+
2
⋀
119895=1
120572119894119895(119905) 119892119895(119906119895(119905 minus 120591119895(119905) 119909))
+
2
⋁
119895=1
120573119894119895(119905) 119892119895(119906119895(119905 minus 120591119895(119905) 119909))
+
2
⋀
119895=1
119879119894119895(119905) 119907119895(119905) +
2
⋁
119895=1
119867119894119895(119905) 119907119895(119905)
119905 = 119905119896 119909 isin Ω
119906119894(119905+
119896 119909) = (1 minus 120574
119894119896) 119906119894(119905minus
119896 119909)
119905 = 119905119896 119896 isin Z
+ 119909 isin Ω
120597119906119894(119905 119909)
120597119899= 0 119905 ge 119905
0 119909 isin 120597Ω
119906119894(1199050+ 119904 119909) = 120595
119894(119904 119909)
minus 120591119895le 119904 le 0 119909 isin Ω
(44)
where 119894 = 1 2 1198881(119905) = 26 119888
2(119905) = 208 119886
11(119905) = minus1 minus cos(119905)
11988612(119905) = 1 + cos(119905) 119886
21(119905) = 1 + sin(119905) 119886
22(119905) = minus1 minus sin(119905)
1198631= 8119863
2= 4 120597119906
119894(119905 119909)120597119899 = 0 (119905 ge 119905
0 119909 = 0 2120587) 120574
1119896= 04
1205742119896
= 02 1205951(sdot) = 120595
1(sdot) = 5 119887
11(119905) = 119887
21(119905) = cos(119905) 119887
12(119905) =
11988722(119905) = minus cos(119905) 119869
1(119905) = 119869
2(119905) = 1 119867
11(119905) = 119867
21(119905) = sin(119905)
11986712(119905) = 119867
22(119905) = minus1 + sin(119905) 119879
11(119905) = 119879
21(119905) = minus sin(119905)
11987912(119905) = 119879
22(119905) = 2 + sin(119905) 120591
1(119905) = 120591
2(119905) = 1 119891
119895(119906119895) =
119906119895(119905 119909) (119895 = 1 2) 119892
119895(119906119895(119905 minus 1 119909)) = 119906
119895(119905 minus 1 119909)119890
minus119906119895(119905minus1119909)
(119895 =
1 2) 12057211(119905) = minus128 120572
21(119905) = 120572
12(119905) = minus1 + cos(119905) 120572
22(119905) =
minus10 12057311(119905) = 128 120573
12(119905) = minus1 + sin(119905) = 120573
21(119905) 12057322(119905) =
8 Abstract and Applied Analysis
6
4
2
0minus5
05
1015
20 76
54
32
10
1199062
119909
119905
5 43
21
119909
Figure 3 State response 1199062(119905 119909) of model (44) without impulsiveeffects
6
4
2
0minus5
05
1015
20 76
54
32
10
1199062
119909
119905
5 43
21
119909
Figure 4 State response 1199062(119905 119909) of model (44) with impulsiveeffects
10 119907119895(119905) = sin(119905) We assume that there exists 119902 = 6 such
that 119905119896+ 2120587 = 119905
119896+119902 Obviously 119891
1 1198912 1198921 and 119892
2satisfy the
assumption (H1) with 1198651= 1198652= 1198661= 1198662= 1 and (H2) and
(H3) are satisfied with a common positive period 2120587
119862 minus (119860 + 120572 + 120573)119865 =[[
[
2
50
04
5
]]
]
(45)
is a nonsingular 119872-matrix The conditions of Theorem 6 aresatisfied hence there exists exactly one 2120596-periodic solutionof the model and all other solutions of the model convergeexponentially to it as 119905 rarr +infin Furthermore the exponentialconverging index can be calculated as120582 = 0021 because here120578119896= 1 and 120578 = 0 The simulation results are shown in Figures
1 2 3 and 4 respectively
5 Conclusions
In this paper periodicity and global exponential stability ofa class of FIRDDCNN model with variable both coefficients
and delays have been investigated By using Halanayrsquos delaydifferential inequality 119872-matrix theory and analytic meth-ods some new sufficient conditions have been established toguarantee the existence uniqueness and global exponentialstability of the periodic solution Moreover the exponentialconvergence rate index can be estimated An example andits simulation have been given to show the effectiveness ofthe obtained results In particular the differentiability ofthe time-varying delays has been removed The dynamicbehaviors of fuzzy neural networks with the property ofexponential periodicity are of great importance inmany areassuch as learning systems
Acknowledgments
The authors would like to thank the editors and the anony-mous reviewers for their valuable comments and construc-tive suggestions This research is supported by the NaturalScience Foundation of Guangxi Autonomous Region (no2012GXNSFBA053003) the National Natural Science Foun-dations of China (60973048 61272077 60974025 60673101and 60939003) National 863 Plan Project (2008 AA04Z4012009 AA043404) the Natural Science Foundation of Shan-dong Province (no Y2007G30) the Scientific andTechnolog-ical Project of Shandong Province (no 2007GG3WZ04016)the Science Foundation of Harbin Institute of Technology(Weihai) (HIT (WH) 200807) theNatural ScientificResearchInnovation Foundation in Harbin Institute of Technology(HIT NSRIF 2001120) the China Postdoctoral ScienceFoundation (2010048 1000) and the Shandong ProvincialKey Laboratory of Industrial Control Technique (QingdaoUniversity)
References
[1] LOChua andL Yang ldquoCellular neural networks theoryrdquo IEEETransactions on Circuits and Systems vol 35 no 10 pp 1257ndash1272 1988
[2] L O Chua and L Yang ldquoCellular neural networks applica-tionsrdquo IEEE Transactions on Circuits and Systems vol 35 no10 pp 1273ndash1290 1988
[3] T Roska and L O Chua ldquoCellular neural networks withnon-linear and delay-type template elements and non-uniformgridsrdquo International Journal of Circuit Theory and Applicationsvol 20 no 5 pp 469ndash481 1992
[4] T Yang L B Yang C W Wu and L O Chua ldquoFuzzycellular neural networks theoryrdquo in Proceedings of the 4th IEEEInternational Workshop on Cellular Neural Networks and TheirApplications (CNNA rsquo96) pp 181ndash186 June 1996
[5] T Yang and L-B Yang ldquoThe global stability of fuzzy cellularneural networkrdquo IEEE Transactions on Circuits and Systems Ivol 43 no 10 pp 880ndash883 1996
[6] T Yang L B Yang C W Wu and L O Chua ldquoFuzzy cellularneural networks applicationsrdquo in Proceedings of the 4th IEEEInternational Workshop on Cellular Neural Networks and TheirApplications (CNNA rsquo96) pp 225ndash230 June 1996
[7] S T Wang K F L Chung and F Duan ldquoApplying theimproved fuzzy cellular neural network IFCNN to white bloodcell detectionrdquo Neurocomputing vol 70 no 7ndash9 pp 1348ndash13592007
Abstract and Applied Analysis 9
[8] T G Barbounis and J B Theocharis ldquoA locally recurrent fuzzyneural network with application to the wind speed predictionusing spatial correlationrdquo Neurocomputing vol 70 no 7ndash9 pp1525ndash1542 2007
[9] H Zhao ldquoGlobal exponential stability and periodicity of cellularneural networks with variable delaysrdquoPhysics Letters A vol 336no 4-5 pp 331ndash341 2005
[10] S Townley A Ilchmann M G Weiss et al ldquoExistence andlearning of oscillations in recurrent neural networksrdquo IEEETransactions on Neural Networks vol 11 no 1 pp 205ndash2142000
[11] Z Huang and Y Xia ldquoExponential periodic attractor ofimpulsive BAM networks with finite distributed delaysrdquo ChaosSolitons amp Fractals vol 39 no 1 pp 373ndash384 2009
[12] Y Liu and W Tang ldquoExponential stability of fuzzy cellularneural networkswith constant and time-varying delaysrdquoPhysicsLetters A vol 323 no 3-4 pp 224ndash233 2004
[13] K Yuan J Cao and J Deng ldquoExponential stability and periodicsolutions of fuzzy cellular neural networks with time-varyingdelaysrdquoNeurocomputing vol 69 no 13ndash15 pp 1619ndash1627 2006
[14] T Huang ldquoExponential stability of fuzzy cellular neural net-works with distributed delayrdquo Physics Letters A vol 351 no 1-2pp 48ndash52 2006
[15] G Bao S Wen and Z Zeng ldquoRobust stability analysis of inter-val fuzzy Cohen-Grossberg neural networks with piecewiseconstant argument of generalized typerdquo Neural Networks vol33 pp 32ndash41 2012
[16] H Zhao N Ding and L Chen ldquoAlmost sure exponential sta-bility of stochastic fuzzy cellular neural networks with delaysrdquoChaos Solitons amp Fractals vol 40 no 4 pp 1653ndash1659 2009
[17] W Ding M Han and M Li ldquoExponential lag synchronizationof delayed fuzzy cellular neural networkswith impulsesrdquoPhysicsLetters A vol 373 no 8-9 pp 832ndash837 2009
[18] L Chen and H Zhao ldquoStability analysis of stochastic fuzzycellular neural networks with delaysrdquo Neurocomputing vol 72no 1ndash3 pp 436ndash444 2008
[19] T Huang ldquoExponential stability of delayed fuzzy cellular neuralnetworks with diffusionrdquo Chaos Solitons amp Fractals vol 31 no3 pp 658ndash664 2007
[20] J Wang and J G Lu ldquoGlobal exponential stability of fuzzycellular neural networks with delays and reaction-diffusiontermsrdquo Chaos Solitons amp Fractals vol 38 no 3 pp 878ndash8852008
[21] Q Song and Z Wang ldquoDynamical behaviors of fuzzy reaction-diffusion periodic cellular neural networks with variable coeffi-cients and delaysrdquo Applied Mathematical Modelling vol 33 no9 pp 3533ndash3545 2009
[22] C Wang Y kao and G Yang ldquoExponential stability ofimpulsive stochastic fuzzy reaction-diffusion Cohen-Grossbergneural networks with mixed delaysrdquo Neurocomputing vol 89pp 55ndash63 2012
[23] T Serrano-Gotarredona and B Linares-Barranco ldquoLog-Domain Implementation of Complex Dynamics Reaction-Diffusion Neural Networksrdquo IEEE Transactions on NeuralNetworks vol 14 no 5 pp 1337ndash1355 2003
[24] J Liang and J Cao ldquoGlobal exponential stability of reaction-diffusion recurrent neural networks with time-varying delaysrdquoPhysics Letters A vol 314 no 5-6 pp 434ndash442 2003
[25] L Wang and D Xu ldquoAsymptotic behavior of a class of reaction-diffusion equations with delaysrdquo Journal of Mathematical Anal-ysis and Applications vol 281 no 2 pp 439ndash453 2003
[26] L Wang and D Xu ldquoGlobal exponential stability of Hopfieldreaction-diffusion neural networks with time-varying delaysrdquoScience in China Series F vol 46 no 6 pp 466ndash474 2003
[27] Z Yang and D Xu ldquoGlobal dynamics for non-autonomousreaction-diffusion neural networks with time-varying delaysrdquoTheoretical Computer Science vol 403 no 1 pp 3ndash10 2008
[28] Y G Kao C C Gao and D K Wang ldquoGlobal exponentialstability of reaction-diffusion Hopfield neural networks withcontinuously distributed delaysrdquo Mathematica Applicata vol21 no 3 pp 457ndash462 2008 (Chinese)
[29] Y-G Kao J-F Guo C-H Wang and X-Q Sun ldquoDelay-dependent robust exponential stability of Markovian jump-ing reaction-diffusion Cohen-Grossberg neural networks withmixed delaysrdquo Journal of the Franklin Institute vol 349 no 6pp 1972ndash1988 2012
[30] H Zhao and Z Mao ldquoBoundedness and stability of nonau-tonomous cellular neural networks with reaction-diffusiontermsrdquo Mathematics and Computers in Simulation vol 79 no5 pp 1603ndash1617 2009
[31] H Zhao and G Wang ldquoExistence of periodic oscillatorysolution of reaction-diffusion neural networks with delaysrdquoPhysics Letters A vol 343 no 5 pp 372ndash383 2005
[32] Q Gan ldquoExponential synchronization of stochastic Cohen-Grossberg neural networks withmixed time-varying delays andreaction-diffusion via periodically intermittent controlrdquoNeuralNetworks vol 31 pp 12ndash21 2012
[33] V Lakshmikantham D D Baınov and P S Simeonov Theoryof Impulsive Differential Equations vol 6 of Series in ModernApplied Mathematics World Scientific Teaneck NJ USA 1989
[34] D D Bainov and P S Simeonov Impulsive Differential Equa-tions Periodic Solutions and Applications Longman HarlowUK 1993
[35] X Song X Xin andWHuang ldquoExponential stability of delayedand impulsive cellular neural networks with partially Lipschitzcontinuous activation functionsrdquo Neural Networks vol 29-30pp 80ndash90 2012
[36] Z Yang and D Xu ldquoExistence and exponential stability ofperiodic solution for impulsive delay differential equations andapplicationsrdquo Nonlinear Analysis Theory Methods amp Applica-tions vol 64 no 1 pp 130ndash145 2006
[37] X Yang X Liao D J Evans and Y Tang ldquoExistence andstability of periodic solution in impulsive Hopfield neuralnetworks with finite distributed delaysrdquo Physics Letters A vol343 no 1ndash3 pp 108ndash116 2005
[38] Y Li and L Lu ldquoGlobal exponential stability and existenceof periodic solution of Hopfield-type neural networks withimpulsesrdquo Physics Letters A vol 333 no 1-2 pp 62ndash71 2004
[39] Y Li L Zhu andP Liu ldquoExistence and stability of periodic solu-tions of delayed cellular neural networksrdquo Nonlinear AnalysisReal World Applications vol 7 no 2 pp 225ndash234 2006
[40] Z H Guan and G Chen ldquoOn delayed impulsive Hopfieldneural networksrdquo Neural Networks vol 12 no 2 pp 273ndash2801999
[41] Z H Guan J Lam andG Chen ldquoOn impulsive autoassociativeneural networksrdquo Neural Networks vol 13 no 1 pp 63ndash692000
[42] W Ding ldquoSynchronization of delayed fuzzy cellular neuralnetworks with impulsive effectsrdquo Communications in NonlinearScience and Numerical Simulation vol 14 no 11 pp 3945ndash39522009
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
8 Abstract and Applied Analysis
6
4
2
0minus5
05
1015
20 76
54
32
10
1199062
119909
119905
5 43
21
119909
Figure 3 State response 1199062(119905 119909) of model (44) without impulsiveeffects
6
4
2
0minus5
05
1015
20 76
54
32
10
1199062
119909
119905
5 43
21
119909
Figure 4 State response 1199062(119905 119909) of model (44) with impulsiveeffects
10 119907119895(119905) = sin(119905) We assume that there exists 119902 = 6 such
that 119905119896+ 2120587 = 119905
119896+119902 Obviously 119891
1 1198912 1198921 and 119892
2satisfy the
assumption (H1) with 1198651= 1198652= 1198661= 1198662= 1 and (H2) and
(H3) are satisfied with a common positive period 2120587
119862 minus (119860 + 120572 + 120573)119865 =[[
[
2
50
04
5
]]
]
(45)
is a nonsingular 119872-matrix The conditions of Theorem 6 aresatisfied hence there exists exactly one 2120596-periodic solutionof the model and all other solutions of the model convergeexponentially to it as 119905 rarr +infin Furthermore the exponentialconverging index can be calculated as120582 = 0021 because here120578119896= 1 and 120578 = 0 The simulation results are shown in Figures
1 2 3 and 4 respectively
5 Conclusions
In this paper periodicity and global exponential stability ofa class of FIRDDCNN model with variable both coefficients
and delays have been investigated By using Halanayrsquos delaydifferential inequality 119872-matrix theory and analytic meth-ods some new sufficient conditions have been established toguarantee the existence uniqueness and global exponentialstability of the periodic solution Moreover the exponentialconvergence rate index can be estimated An example andits simulation have been given to show the effectiveness ofthe obtained results In particular the differentiability ofthe time-varying delays has been removed The dynamicbehaviors of fuzzy neural networks with the property ofexponential periodicity are of great importance inmany areassuch as learning systems
Acknowledgments
The authors would like to thank the editors and the anony-mous reviewers for their valuable comments and construc-tive suggestions This research is supported by the NaturalScience Foundation of Guangxi Autonomous Region (no2012GXNSFBA053003) the National Natural Science Foun-dations of China (60973048 61272077 60974025 60673101and 60939003) National 863 Plan Project (2008 AA04Z4012009 AA043404) the Natural Science Foundation of Shan-dong Province (no Y2007G30) the Scientific andTechnolog-ical Project of Shandong Province (no 2007GG3WZ04016)the Science Foundation of Harbin Institute of Technology(Weihai) (HIT (WH) 200807) theNatural ScientificResearchInnovation Foundation in Harbin Institute of Technology(HIT NSRIF 2001120) the China Postdoctoral ScienceFoundation (2010048 1000) and the Shandong ProvincialKey Laboratory of Industrial Control Technique (QingdaoUniversity)
References
[1] LOChua andL Yang ldquoCellular neural networks theoryrdquo IEEETransactions on Circuits and Systems vol 35 no 10 pp 1257ndash1272 1988
[2] L O Chua and L Yang ldquoCellular neural networks applica-tionsrdquo IEEE Transactions on Circuits and Systems vol 35 no10 pp 1273ndash1290 1988
[3] T Roska and L O Chua ldquoCellular neural networks withnon-linear and delay-type template elements and non-uniformgridsrdquo International Journal of Circuit Theory and Applicationsvol 20 no 5 pp 469ndash481 1992
[4] T Yang L B Yang C W Wu and L O Chua ldquoFuzzycellular neural networks theoryrdquo in Proceedings of the 4th IEEEInternational Workshop on Cellular Neural Networks and TheirApplications (CNNA rsquo96) pp 181ndash186 June 1996
[5] T Yang and L-B Yang ldquoThe global stability of fuzzy cellularneural networkrdquo IEEE Transactions on Circuits and Systems Ivol 43 no 10 pp 880ndash883 1996
[6] T Yang L B Yang C W Wu and L O Chua ldquoFuzzy cellularneural networks applicationsrdquo in Proceedings of the 4th IEEEInternational Workshop on Cellular Neural Networks and TheirApplications (CNNA rsquo96) pp 225ndash230 June 1996
[7] S T Wang K F L Chung and F Duan ldquoApplying theimproved fuzzy cellular neural network IFCNN to white bloodcell detectionrdquo Neurocomputing vol 70 no 7ndash9 pp 1348ndash13592007
Abstract and Applied Analysis 9
[8] T G Barbounis and J B Theocharis ldquoA locally recurrent fuzzyneural network with application to the wind speed predictionusing spatial correlationrdquo Neurocomputing vol 70 no 7ndash9 pp1525ndash1542 2007
[9] H Zhao ldquoGlobal exponential stability and periodicity of cellularneural networks with variable delaysrdquoPhysics Letters A vol 336no 4-5 pp 331ndash341 2005
[10] S Townley A Ilchmann M G Weiss et al ldquoExistence andlearning of oscillations in recurrent neural networksrdquo IEEETransactions on Neural Networks vol 11 no 1 pp 205ndash2142000
[11] Z Huang and Y Xia ldquoExponential periodic attractor ofimpulsive BAM networks with finite distributed delaysrdquo ChaosSolitons amp Fractals vol 39 no 1 pp 373ndash384 2009
[12] Y Liu and W Tang ldquoExponential stability of fuzzy cellularneural networkswith constant and time-varying delaysrdquoPhysicsLetters A vol 323 no 3-4 pp 224ndash233 2004
[13] K Yuan J Cao and J Deng ldquoExponential stability and periodicsolutions of fuzzy cellular neural networks with time-varyingdelaysrdquoNeurocomputing vol 69 no 13ndash15 pp 1619ndash1627 2006
[14] T Huang ldquoExponential stability of fuzzy cellular neural net-works with distributed delayrdquo Physics Letters A vol 351 no 1-2pp 48ndash52 2006
[15] G Bao S Wen and Z Zeng ldquoRobust stability analysis of inter-val fuzzy Cohen-Grossberg neural networks with piecewiseconstant argument of generalized typerdquo Neural Networks vol33 pp 32ndash41 2012
[16] H Zhao N Ding and L Chen ldquoAlmost sure exponential sta-bility of stochastic fuzzy cellular neural networks with delaysrdquoChaos Solitons amp Fractals vol 40 no 4 pp 1653ndash1659 2009
[17] W Ding M Han and M Li ldquoExponential lag synchronizationof delayed fuzzy cellular neural networkswith impulsesrdquoPhysicsLetters A vol 373 no 8-9 pp 832ndash837 2009
[18] L Chen and H Zhao ldquoStability analysis of stochastic fuzzycellular neural networks with delaysrdquo Neurocomputing vol 72no 1ndash3 pp 436ndash444 2008
[19] T Huang ldquoExponential stability of delayed fuzzy cellular neuralnetworks with diffusionrdquo Chaos Solitons amp Fractals vol 31 no3 pp 658ndash664 2007
[20] J Wang and J G Lu ldquoGlobal exponential stability of fuzzycellular neural networks with delays and reaction-diffusiontermsrdquo Chaos Solitons amp Fractals vol 38 no 3 pp 878ndash8852008
[21] Q Song and Z Wang ldquoDynamical behaviors of fuzzy reaction-diffusion periodic cellular neural networks with variable coeffi-cients and delaysrdquo Applied Mathematical Modelling vol 33 no9 pp 3533ndash3545 2009
[22] C Wang Y kao and G Yang ldquoExponential stability ofimpulsive stochastic fuzzy reaction-diffusion Cohen-Grossbergneural networks with mixed delaysrdquo Neurocomputing vol 89pp 55ndash63 2012
[23] T Serrano-Gotarredona and B Linares-Barranco ldquoLog-Domain Implementation of Complex Dynamics Reaction-Diffusion Neural Networksrdquo IEEE Transactions on NeuralNetworks vol 14 no 5 pp 1337ndash1355 2003
[24] J Liang and J Cao ldquoGlobal exponential stability of reaction-diffusion recurrent neural networks with time-varying delaysrdquoPhysics Letters A vol 314 no 5-6 pp 434ndash442 2003
[25] L Wang and D Xu ldquoAsymptotic behavior of a class of reaction-diffusion equations with delaysrdquo Journal of Mathematical Anal-ysis and Applications vol 281 no 2 pp 439ndash453 2003
[26] L Wang and D Xu ldquoGlobal exponential stability of Hopfieldreaction-diffusion neural networks with time-varying delaysrdquoScience in China Series F vol 46 no 6 pp 466ndash474 2003
[27] Z Yang and D Xu ldquoGlobal dynamics for non-autonomousreaction-diffusion neural networks with time-varying delaysrdquoTheoretical Computer Science vol 403 no 1 pp 3ndash10 2008
[28] Y G Kao C C Gao and D K Wang ldquoGlobal exponentialstability of reaction-diffusion Hopfield neural networks withcontinuously distributed delaysrdquo Mathematica Applicata vol21 no 3 pp 457ndash462 2008 (Chinese)
[29] Y-G Kao J-F Guo C-H Wang and X-Q Sun ldquoDelay-dependent robust exponential stability of Markovian jump-ing reaction-diffusion Cohen-Grossberg neural networks withmixed delaysrdquo Journal of the Franklin Institute vol 349 no 6pp 1972ndash1988 2012
[30] H Zhao and Z Mao ldquoBoundedness and stability of nonau-tonomous cellular neural networks with reaction-diffusiontermsrdquo Mathematics and Computers in Simulation vol 79 no5 pp 1603ndash1617 2009
[31] H Zhao and G Wang ldquoExistence of periodic oscillatorysolution of reaction-diffusion neural networks with delaysrdquoPhysics Letters A vol 343 no 5 pp 372ndash383 2005
[32] Q Gan ldquoExponential synchronization of stochastic Cohen-Grossberg neural networks withmixed time-varying delays andreaction-diffusion via periodically intermittent controlrdquoNeuralNetworks vol 31 pp 12ndash21 2012
[33] V Lakshmikantham D D Baınov and P S Simeonov Theoryof Impulsive Differential Equations vol 6 of Series in ModernApplied Mathematics World Scientific Teaneck NJ USA 1989
[34] D D Bainov and P S Simeonov Impulsive Differential Equa-tions Periodic Solutions and Applications Longman HarlowUK 1993
[35] X Song X Xin andWHuang ldquoExponential stability of delayedand impulsive cellular neural networks with partially Lipschitzcontinuous activation functionsrdquo Neural Networks vol 29-30pp 80ndash90 2012
[36] Z Yang and D Xu ldquoExistence and exponential stability ofperiodic solution for impulsive delay differential equations andapplicationsrdquo Nonlinear Analysis Theory Methods amp Applica-tions vol 64 no 1 pp 130ndash145 2006
[37] X Yang X Liao D J Evans and Y Tang ldquoExistence andstability of periodic solution in impulsive Hopfield neuralnetworks with finite distributed delaysrdquo Physics Letters A vol343 no 1ndash3 pp 108ndash116 2005
[38] Y Li and L Lu ldquoGlobal exponential stability and existenceof periodic solution of Hopfield-type neural networks withimpulsesrdquo Physics Letters A vol 333 no 1-2 pp 62ndash71 2004
[39] Y Li L Zhu andP Liu ldquoExistence and stability of periodic solu-tions of delayed cellular neural networksrdquo Nonlinear AnalysisReal World Applications vol 7 no 2 pp 225ndash234 2006
[40] Z H Guan and G Chen ldquoOn delayed impulsive Hopfieldneural networksrdquo Neural Networks vol 12 no 2 pp 273ndash2801999
[41] Z H Guan J Lam andG Chen ldquoOn impulsive autoassociativeneural networksrdquo Neural Networks vol 13 no 1 pp 63ndash692000
[42] W Ding ldquoSynchronization of delayed fuzzy cellular neuralnetworks with impulsive effectsrdquo Communications in NonlinearScience and Numerical Simulation vol 14 no 11 pp 3945ndash39522009
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
Abstract and Applied Analysis 9
[8] T G Barbounis and J B Theocharis ldquoA locally recurrent fuzzyneural network with application to the wind speed predictionusing spatial correlationrdquo Neurocomputing vol 70 no 7ndash9 pp1525ndash1542 2007
[9] H Zhao ldquoGlobal exponential stability and periodicity of cellularneural networks with variable delaysrdquoPhysics Letters A vol 336no 4-5 pp 331ndash341 2005
[10] S Townley A Ilchmann M G Weiss et al ldquoExistence andlearning of oscillations in recurrent neural networksrdquo IEEETransactions on Neural Networks vol 11 no 1 pp 205ndash2142000
[11] Z Huang and Y Xia ldquoExponential periodic attractor ofimpulsive BAM networks with finite distributed delaysrdquo ChaosSolitons amp Fractals vol 39 no 1 pp 373ndash384 2009
[12] Y Liu and W Tang ldquoExponential stability of fuzzy cellularneural networkswith constant and time-varying delaysrdquoPhysicsLetters A vol 323 no 3-4 pp 224ndash233 2004
[13] K Yuan J Cao and J Deng ldquoExponential stability and periodicsolutions of fuzzy cellular neural networks with time-varyingdelaysrdquoNeurocomputing vol 69 no 13ndash15 pp 1619ndash1627 2006
[14] T Huang ldquoExponential stability of fuzzy cellular neural net-works with distributed delayrdquo Physics Letters A vol 351 no 1-2pp 48ndash52 2006
[15] G Bao S Wen and Z Zeng ldquoRobust stability analysis of inter-val fuzzy Cohen-Grossberg neural networks with piecewiseconstant argument of generalized typerdquo Neural Networks vol33 pp 32ndash41 2012
[16] H Zhao N Ding and L Chen ldquoAlmost sure exponential sta-bility of stochastic fuzzy cellular neural networks with delaysrdquoChaos Solitons amp Fractals vol 40 no 4 pp 1653ndash1659 2009
[17] W Ding M Han and M Li ldquoExponential lag synchronizationof delayed fuzzy cellular neural networkswith impulsesrdquoPhysicsLetters A vol 373 no 8-9 pp 832ndash837 2009
[18] L Chen and H Zhao ldquoStability analysis of stochastic fuzzycellular neural networks with delaysrdquo Neurocomputing vol 72no 1ndash3 pp 436ndash444 2008
[19] T Huang ldquoExponential stability of delayed fuzzy cellular neuralnetworks with diffusionrdquo Chaos Solitons amp Fractals vol 31 no3 pp 658ndash664 2007
[20] J Wang and J G Lu ldquoGlobal exponential stability of fuzzycellular neural networks with delays and reaction-diffusiontermsrdquo Chaos Solitons amp Fractals vol 38 no 3 pp 878ndash8852008
[21] Q Song and Z Wang ldquoDynamical behaviors of fuzzy reaction-diffusion periodic cellular neural networks with variable coeffi-cients and delaysrdquo Applied Mathematical Modelling vol 33 no9 pp 3533ndash3545 2009
[22] C Wang Y kao and G Yang ldquoExponential stability ofimpulsive stochastic fuzzy reaction-diffusion Cohen-Grossbergneural networks with mixed delaysrdquo Neurocomputing vol 89pp 55ndash63 2012
[23] T Serrano-Gotarredona and B Linares-Barranco ldquoLog-Domain Implementation of Complex Dynamics Reaction-Diffusion Neural Networksrdquo IEEE Transactions on NeuralNetworks vol 14 no 5 pp 1337ndash1355 2003
[24] J Liang and J Cao ldquoGlobal exponential stability of reaction-diffusion recurrent neural networks with time-varying delaysrdquoPhysics Letters A vol 314 no 5-6 pp 434ndash442 2003
[25] L Wang and D Xu ldquoAsymptotic behavior of a class of reaction-diffusion equations with delaysrdquo Journal of Mathematical Anal-ysis and Applications vol 281 no 2 pp 439ndash453 2003
[26] L Wang and D Xu ldquoGlobal exponential stability of Hopfieldreaction-diffusion neural networks with time-varying delaysrdquoScience in China Series F vol 46 no 6 pp 466ndash474 2003
[27] Z Yang and D Xu ldquoGlobal dynamics for non-autonomousreaction-diffusion neural networks with time-varying delaysrdquoTheoretical Computer Science vol 403 no 1 pp 3ndash10 2008
[28] Y G Kao C C Gao and D K Wang ldquoGlobal exponentialstability of reaction-diffusion Hopfield neural networks withcontinuously distributed delaysrdquo Mathematica Applicata vol21 no 3 pp 457ndash462 2008 (Chinese)
[29] Y-G Kao J-F Guo C-H Wang and X-Q Sun ldquoDelay-dependent robust exponential stability of Markovian jump-ing reaction-diffusion Cohen-Grossberg neural networks withmixed delaysrdquo Journal of the Franklin Institute vol 349 no 6pp 1972ndash1988 2012
[30] H Zhao and Z Mao ldquoBoundedness and stability of nonau-tonomous cellular neural networks with reaction-diffusiontermsrdquo Mathematics and Computers in Simulation vol 79 no5 pp 1603ndash1617 2009
[31] H Zhao and G Wang ldquoExistence of periodic oscillatorysolution of reaction-diffusion neural networks with delaysrdquoPhysics Letters A vol 343 no 5 pp 372ndash383 2005
[32] Q Gan ldquoExponential synchronization of stochastic Cohen-Grossberg neural networks withmixed time-varying delays andreaction-diffusion via periodically intermittent controlrdquoNeuralNetworks vol 31 pp 12ndash21 2012
[33] V Lakshmikantham D D Baınov and P S Simeonov Theoryof Impulsive Differential Equations vol 6 of Series in ModernApplied Mathematics World Scientific Teaneck NJ USA 1989
[34] D D Bainov and P S Simeonov Impulsive Differential Equa-tions Periodic Solutions and Applications Longman HarlowUK 1993
[35] X Song X Xin andWHuang ldquoExponential stability of delayedand impulsive cellular neural networks with partially Lipschitzcontinuous activation functionsrdquo Neural Networks vol 29-30pp 80ndash90 2012
[36] Z Yang and D Xu ldquoExistence and exponential stability ofperiodic solution for impulsive delay differential equations andapplicationsrdquo Nonlinear Analysis Theory Methods amp Applica-tions vol 64 no 1 pp 130ndash145 2006
[37] X Yang X Liao D J Evans and Y Tang ldquoExistence andstability of periodic solution in impulsive Hopfield neuralnetworks with finite distributed delaysrdquo Physics Letters A vol343 no 1ndash3 pp 108ndash116 2005
[38] Y Li and L Lu ldquoGlobal exponential stability and existenceof periodic solution of Hopfield-type neural networks withimpulsesrdquo Physics Letters A vol 333 no 1-2 pp 62ndash71 2004
[39] Y Li L Zhu andP Liu ldquoExistence and stability of periodic solu-tions of delayed cellular neural networksrdquo Nonlinear AnalysisReal World Applications vol 7 no 2 pp 225ndash234 2006
[40] Z H Guan and G Chen ldquoOn delayed impulsive Hopfieldneural networksrdquo Neural Networks vol 12 no 2 pp 273ndash2801999
[41] Z H Guan J Lam andG Chen ldquoOn impulsive autoassociativeneural networksrdquo Neural Networks vol 13 no 1 pp 63ndash692000
[42] W Ding ldquoSynchronization of delayed fuzzy cellular neuralnetworks with impulsive effectsrdquo Communications in NonlinearScience and Numerical Simulation vol 14 no 11 pp 3945ndash39522009
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
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
