Volume 135, number 6,7 PHYSICS LETTERS A 6 March 1989

STATIC AND D Y N A M I C SCAL ING LAWS N E A R T H E S Y M M E T R Y - B R E A K I N G C H A O S T R A N S I T I O N IN T H E D O U B L E - W E L L P O T E N T I A L SYSTEM

Akira YAMAGUCHI , Hirokazu FUJISAKA and Masayoshi I N O U E Department of Physics, Kagoshima University, Kagoshima 890, Japan

Received 8 November 1988; accepted for publication 4 January 1989 Communicated by A.P. Fordy

The fluctuation dynamics near the symmetry-breaking chaos transition in the double-well potential system under an external periodic excitation is studied with the fluctuation spectrum theory and the order-q time correlation function. The anomalous elongation of temporal correlation near the transition point, associated with the development of the interminency characteristic, is shown to be described especially by the dynamic scaling law of the order-q time correlation function.

The band-splitting phenomenon is a typical chaos- chaos transition observed in a wide range o f dynam- ical systems [ 1,2]. An important characteristic of this transition is that this brings about an intermittency phenomenon associated with the symmetry change in chaotic states [3] . In a previous paper [4] , we have shown that the symmetry-breaking transition is observed in the particle mot ion in a double-well po- tential under a periodic excitation ~. The main aim of the present note is to study its symmetry dynam- ics in the vicinity o f the transition point by utilizing the fluctuation spectrum theory [6 ] and the order- q time correlation function approach [ 7 ] to a steady time series.

The model equation is written as

d U ( x ) £ ( t ) = - y ~ - - - + Fcos(og~t ) , (1)

dx

where x ( t ) is the particle position at time t and 7 is the damping rate. F and o9~ are the amplitude and the angular frequency of the external excitation, re- spectively. The potential U(x) has the form

U(x) = ½ax2 + ¼bx 4 , (2)

where a ( < 0 ) and b ( > 0 ) are constants, is sym-

~ l For the double-well potential system, see, e.g., references cited in ref. [ 5 ].

metric under space inversion, U ( - x ) = U(x) , and has two minima at x± = +_x/-Z~/b. In the present note F is chosen as the control parameter and other parameter values are set as a = - 10, b = 100, 7= 1 and o9e=3.5 [8]. As was previously reported [4] , the symmetry-breaking chaos transition is observed at F=F, (~0.84925) as F is gradually decreased from above F, . For F ~ F , the particle eventually stays in one of wells. It depends on the initial con- dition whether the particle is in the right well or in the left well.

Fig. 1 shows the typical temporal evolutions of the particle position for three values o fF . For F>F, the particle moves around over two regions x < 0 and x > 0 . As the strength F o f the external force ap- proaches F , , the average duration in one of the wells becomes long. This characteristic is called the inter- mittency associated with the symmetry change [ 9,10 ]. For F~< F, , the particle position eventually lies in one o f the wells. In fig. 2, typical phase por- traits near the threshold F , are drawn.

In order to study the development of the inter- mittency for F- ,F , we introduce a time series of the symmetry variable, which is defined as follows. Let t ~ - n T ( n = 0 , 1, 2, ...), be the discrete times where T= 2rt/oge is the period of the external force, x(tn) being the particle position at time t~. The symmetry variable is defined as [3,4]

320 0375-9601/89/$ 03.50 © Elsevier Science Publishers B.V. (Nor th-Hol land Physics Publishing Divis ion)

Volume 135, number 6,7 PHYSICS LETTERS A

o,~ ~, ~ N I l V ~ ~ ~ ~,~ ~-ii:, i;~ o 0.4 .

(b ) 0

-0,4"

6 March 1989

0.4" Xl

(c) o

-0.4-

2~0 400 61:)0 8bO 1~00 n

Fig. 1. Typical temporal evolutions of the particle position near the threshold F, . The F value is 0.860 (a), 0.851 (b) and 0.849 (c). One observes the intermittency characteristic for slightly above F, . In the present paper, numerical integrations were carried out with the Runge-Kutta method with the time increment At= (2n/o9~)/90=0.019946 ....

n -0.5

0.5

.I X

0.5

t -0.5 0

- 0 . 5 .

Fig. 2. Phase portraits for (a) F = 0.865 ( > F , ), and (b) F = 0.849 ( < F , ). There is an equivalent attractor in the region x < 0 for F~< F,.

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Volume 135, number 6,7 PHYSICS LETTERS A 6 March 1989

u , , = + 1 , x(t, ,) > 0 ,

= - 1 , x ( t , ) < 0 . (3)

The t ime series

{u,,} = Uo, u,, u2 .... (4)

is deterministically generated because x ( t ) obeys the determinist ic equat ion of mot ion (1) . For F>~F,, the t ime series consists o f two values, + 1 and - 1. However, for F>F,, depending on the initial con- dition, y , eventually takes either + 1 or - 1 for any n. For F slightly above F , , once the particle is in one well, it takes a long t ime to migrate to the other.

The global fluctuation characteristic of { u,} can be studied with the characteristic function [ 11 ]

2 q = - I l i m l l n M q ( n ) (d2q/dq>~0) , (5) q n ~ n

with

Mq(n) - (exp(qno~,)) , (6)

where

o~n = - uj (7) /7/=0

is the local average of uj, ( ) being the ensemble av- erage. For F~F. , a. takes + 1 or - 1, independently of n, i.e., 2q= + 1 or - I, independently ofq. On the other hand, for F>F., a. can take various values between - 1 and + 1 for large n. By assuming that p. (o~), the probabil i ty density that c~. takes a value between oL and o~ + do~, is asymptotically written as [6]

p,,(o~) ~ e x p [ - a ( a ) n ] , (8)

the fluctuation spectrum a(c¢) is given by the t ransformat ion

d(q2q) cr(o~(q))=q2d2q (9) c ~ ( q ) - dq ' dq '

(do~(q)/dq>~O, d2o'(c~)/dot2>0). The quantit ies defined through

d a ( q ) Z(c~) ~ [a,, ( ~ ) ]_ , 10) Zq - dq '

(Xq=Z(o~(q))) evaluate the intensities of the fluc-

tuations specified with the variables q and a , re- spectively [ 12 ].

I f we put

Mq(n) =Oq(n) exp(q2qn) (11 )

(lim,,~oo n-1 In Qq(n) = 0 ) , Qq(n) describes the ex- plicit temporal correlation embedded in {un}. The poles of

~q((.O)= ~ aq(n) cos(o)n) (12) n = 0

(o) # 0) , thus yield the characteristic frequencies and the damping rates of the characteristic mot ion in {u,} [7]. By studying both )~q and Qq(n), the statistical characterization of o~, becomes complete. Hereafter we will discuss how the development of the inter- mit tency for F ~ F , can be analyzed with the quan- tities given above.

As was previously reported [ 4 ], slightly above F , , J.q obeys the scaling law

2q =L(q /x ) , (13)

where L(x ) is a scaling function. The x evaluates the boundaries among characteristic regions of q and shrinks for F--,F, as

Koc(F-F, ) ~ ( O > 0 ) , (14)

where the numerical calculation suggests ~ = 0.7-0.8 [ 4 ]. The characteristic t ime r relevant to the present intermit tency tends to diverge as ( F - F , ) - ~ . Since roc x - ~ [ 4 ], we get ~ = 7. Recently Grebogi et al. [ 10 ] have obtained 7 = 0.703 by evaluating the expanding and contracting eigenvalues of the unstable periodic orbit. The quantities c~(q) etc. satisfy the scaling relations

o~(q) =A(q/x) ,

~(~) =KS(cr),

1 Yq= ~ A ' ( q l x ) , l [ s , , ( a ) ] - ,

Z ( a ) = x

(15)

(16)

(17)

where A(x)=-d[xL(x ) ] /dx and S ( y ) = A - t ( y ) × [ y - L ( A - 1 (y)) ], x=A - 1 (y) being the inverse function ofA (x) =y. Since 2q is relevant to the global characteristic, eqs. ( 13 ), ( 15 ) - ( 17 ) are the "s tat ic" scaling laws.

The "dynamic" version of the scaling relation is postulated as [ 13 ]

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Volume 135, number 6,7 PHYSICS LETTERS A 6 March 1989

Qq(n) =F(q/x , xg2(q/x)n) , (18)

1G( q ' co ) 3 . ( o 9 ) = 7c \ x Ka-~lx) ' (19)

where ~ ( x ) is the scaling function for the charac- teristic frequency (or damping rate). The scaling functions F and G are related to each other via

G(x, y) ~ ~ F(x, z) cos(yz) d z . (20) 0

Next let us turn to the comparison with numerical results. As was discussed in ref. [13] , even a small numerical error in 2q evaluated with eq. (5) for a large, finite n may bring about an enormous error especially in -~o (°9). So, in order to avoid such trou- ble, we have used the continued fraction expansion method for Mq(n) introduced in ref. [14] . For ex- ample the two-pole approximation determines ,~q and Qq(n) only with Me( 1 ), Mq(2) and Mq(3). The nu- merical results o f 2q for three values o f F are shown in fig. 3a. As F ~ F , + , the slope Of Aq at q = 0 be- comes steep. This is due to the development of the intermittency characteristic as in fig. 1. The ther- modynamic variables are drawn in fig. 4.

The present symmetry-breaking phenomenon is similar to the band-splitting phenomenon typically observed in one-dimensional chaotic dynamics. In ref. [ 13], we have developed the theory o f ,~q and aq(n), based on the two-level stochastic model, where x plays the role o f the transition probability from the + ( - ) state to - ( + ) in a unit step. Ac- cording to this model, the scaling function L is writ- ten as

L ( x ) - ~ - 1 (21) x

This readily leads to

A ( x ) = x ( l + x 2 ) -1/2, A ' ( x ) = ( l + x 2 ) -3/2

(22)

S ( x ) = I - ( 1 - x ) 1/2 , S " ( x ) = ( I - x 2 ) -3 /2

(23)

( A' ( x ) = 1/S" ( A ( x ) ) ). Furthermore the order-q time correlation function Qq(n) (eq. (11 )) is ob- tained as

(a)

0.5.

0.62 0.64 Ci

.-0.5

=T

(D) 1 Aq" -

0.5. S

" l b ~ ° -'0.5

-7 . . . . . . . . . . . . .

Fig. 3. (a) Numerical results Of 2q for F=0.85023 ( ---F~ ), 0.84981 (--F2) and 0.84951 (---F3). In the present paper, the statistical analyses were carried out with the two-pole approximation of the continued fraction expansion for the Laplace transform of Mq(n). For these F values, e=-F-F, is about 9.8X 10 -4, 5.6× 10 -4 and 3.2 × 10 - 4 for F= Fi, F2 and F3, respectively. (b) The scaling form of Aq for F=FI ( A ), F2 ( × ) and F3 ( © ) in (a), where x is de- termined by equating the slope of A# at q= 0 with 1/2x. We find x=7.6X 10 -3, 5.3×10 -3 and 3.6× 10 -3 for F=FI, F2 and F3, respectively. The solid line is the theoretical expression 2 q = L (q/ x) with (21).

Qq(n) =jq(o)_~= j(1 ) e x p ( - y q n ) , (24)

which contains only one characteristic time 1/yq. Therefore

2q(o9) =Jq~ ) exp (?q) - c o s ( c o ) 4[sinh2(yq/2 )+sin2(o9/2 ) ]

~)q (0,)-7~ 0 ) . (25) ~jqtl) 72+o9 2

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Volume 135, number 6,7 PHYSICS LETTERS A 6 March 1989

(a )

o~(q)

-O.b4 -0~ -1

0 O.b2 ' q 0.04

~ o b> ~(o~) 0.004"

2

-1 - ( i5 (

I

0 :5 Ol 1

(C)

-0.04 -0.02 0 0.02 .04 q -1 -0.5 0 0.5 ~ 1

Xq

200

100

Fig. 4. Thermodynamic quantities derived from 2q in fig. 3a. The three solid lines in each figure are results for F=Ft, F2 and F3. As F approaches F., c~ (q) becomes steep, a( a ) tends to be suppressed, and Zq and Z ( ct ) tend to diverge.

The dynamic scaling law is written as

~q=X,.Q( q / x ) . (26)

By evaluating the exact results f rom the stochastic

model in the vicinity of the transition point, the scal- ing functions are obtained as

F ( x , y ) = 1 - K ( x ) + K ( x ) e -y , (27)

(a) 1 o~(c

-'iO -'5

J (b) ~ (a ) ,

o g q/,~ 1'o , ~ o.5i

-1 -d5 0 0.5 6e

(C) 1 ~

-1'o -'s o ~ Cl/+ ~b Fig. 5. The scaling plots of the results in fig. 4. The symbols/x, × and are theoretical results with scaling functions (22) and (23).

0 are results for F=F~, F2 and F3, respectively. The solid curves

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Volume 135, number 6,7 PHYSICS LETTERS A 6 March 1989

t2(x) =2x / l + x z , (28)

K(x) (y~O), (29) G(x, y) = 21x/-i--+~ ( 1 +yZ)

where

1( , ) ,3o, K ( x ) - - ~ 1 - 1 ~ ~ .

The expansion coefficient in eq. (24) satisfies

J(l)=K(q/x) (31) q

In comparing numerical results with those derived from the stochastic model, the characteristic quan-

tity X was determined by equating the slope of nu- mericaUy obtained 2q at q=0 with 1/2K ( =L' (0)/ r ) . Fig. 3b is the scaling plots of 2~'s given in fig. 3a, where the solid curve is the analytic expression (21 ). Fig. 5 shows the scaling forms of the quantities in fig. 4. The solid lines are the scaling functions ( 15 )- ( 17 ) with (22) and (23).

Let us turn to the dynamical aspect. Fig. 6 shows the numerical results of the damping rate ~q and the expansion coefficient J~) in (24). The develop- ment of the intermittency causes the critical slowing- down of fluctuations, which is observed in the de- crease of Yo as F--,F,. Simultaneously the concave

-o.b4 -0:02 o o.b2 o.~ q

-o.~ - o ~ o o~62 o.t~ q

Fig. 6. Numerical results of 7q(a) and J ~ ) (b) for F = FI, F2 and F3. As F-.F,, yq shows a critical slowing down and _~](~) ap- proaches 0.5 for any q ( 4 0 ) .

10 Vq

1

-io -'s o ~ q/,~ io

0.5 j~l,

q

i

-~o -'s o s Cl/K lo

Fig.7. The scaling forms of 7q and jqtl) given in fig. 6. The sym- bols A, X and O correspond to the results for F~, F2 and/73, respectively. The solid lines are analytical results, eqs. (28) and (31 ), obtained from the stochastic model.

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Volume 135, number 6,7 PHYSICS LETTERS A 6 March 1989

structure of l~ t ) becomes remarkable . The scaling v q

proper t ies of~,q and jq~t) are given in fig. 7. The solid

curves are the analyt ic expressions (26) with (28) and (31) with (30) .

According to the above two-level stochastic model, the convent ional double t ime correlat ion funct ion is

calculated as Cn = (UnUo) = exp ( - yon), where the decay constant 70 ( = 2 r ) coincides with limq~o 7q, Y~ being given in eq. (26) . So the power spec t rum is wri t ten as P(a~) ~ 2),o/(72 +o22) for small 7o and 02. This expression has been already shown to be in good agreement with the numerica l result o f the power spect rum (fig. 5 in ref. [4] ). The va l id i ty o f the sim-

ple Lorentz ian for P(og) also suggests the applica- bi l i ty of the single exponent ia l ly decaying te rm as in

(24) . We have calculated Qq (n ) with the three-pole ap-

p rox imat ion of the cont inued fract ion expansion for Mq(n) . For the three-pole approx ima t ion we need the moments Mq(1) , Mq(2) . . . . . Mq(5) . The three- pole approx ima t ion gives an addi t iona l exponen- t ially decaying te rm in eq. (24) . Numer ica l ly we found that the add i t iona l t e rm is less than the J~ ') term, the rat io being at most O ( 1 0 - 3 ) . This also

supports the usefulness o f the two-pole approx ima- t ion o f the cont inued fract ion expansion [ 13 ].

Final ly let us give a commen t on the compar i son

among other scaling funct ions of 2q. In the previous papers [15,3 ], we have proposed two other scaling functions o f 2q in the band-spl i t t ing phenomenon as L FIU (x ) = ( 2 / n ) a r c t a n ( n x / 4 ) and L vFI(X) = (1 /X)

x l n [ c o s h ( x ) ]. These funct ions including (21) co- incide for Ix] << 1 and for Ix[ >> 1 with each other. However in the in te rmedia te region ( [xl ~ 1 ), they

are slightly different f rom each other. It seems that the scaling function (21 ) is closer to the numerica l results than L ~IU and L vFl.

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[ 2 ] P. Collet and J.-P. Eckmann, Iterated maps on the interval as dynamical systems (Birkh~iuser, Basel, 1980).

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[4] H. Ishii, H. Fujisaka and M. Inoue, Phys. Len. A 116 (1986) 257.

[5]Y.H. Kao, J.C. Huang and Y.S. Gou, Phys. Lett. A 131 (1988) 91.

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[ 8 ] W.-H. Steeb, W. Erig and A. Kunick, Phys. Lett. A 93 (1983) 267.

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[ 10] C. Grebogi, E. Ott, F. Romeiras and J.A. Yorke, Phys. Rev. A 36 (1987) 5365, and references therein.

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[12] H. Fujisaka and M. Inoue, Ensemble processing and the global characterization of temporal and spatial fluctuations, Phys. Rev. A 39 (February 1989).

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