Scaling of Lyapunov Exponents of Coupled Chaotic Systems


Scaling of Lyapunov Exponents of Coupled Chaotic Systems
Rudiger Zillmer, Volker Ahlers, and Arkady Pikovsky
Department of Physics, University of Potsdam, Postfach 601553, D-14415 Potsdam, Germany

(January 11, 2000)

Published in Physical Review E 61(1), 332{341 (2000) Copyright 2000 by The American Physical Society

Abstract
We develop a statistical theory of the coupling sensitivity of chaos. The e ect was rst described by Daido Prog. Theor. Phys. 72, 853 (1984)], it appears as a logarithmic singularity in the Lyapunov exponent in coupled chaotic systems at very small couplings. Using a continuous-time stochastic model for the coupled systems we derive a scaling relation for the largest Lyapunov exponent. The singularity is shown to depend on the coupling and the systems' mismatch. Generalizations to the cases of asymmetrical coupling and three interacting oscillators are considered, too. The analytical results are con rmed by numerical simulations. PACS number(s): 05.45.Xt

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I. INTRODUCTION
The dynamics of coupled chaotic systems attracted large interest recently. Many interesting phenomena, in particular di erent kinds of synchronization, can already be observed in simplest cases of two interacting chaotic attractors 1{3]. Whilst the synchronization occurs for couplings large enough to suppress a chaos-induced tendency to desynchronization, an interesting anomality in the dynamics happens for very small couplings already. This is the e ect of coupling sensitivity of chaos, rst observed by Daido 4{7] (see also 8,9]): the dependence of the largest Lyapunov exponent on the coupling parameter " has a singularity 1=j ln "j for small couplings " ! 0. The largest Lyapunov exponent thus increases when weak coupling is introduced. This counterintuitive e ect has been described as a coupling-induced instability 9,10]. The largest Lyapunov exponent measures the growth rate of in nitesimal perturbations to chaotic trajectories and serves as one of the most important characteristics of chaotic motion, in numerics it is a standard tool for proving the existence of chaos. Moreover, many physically relevant properties of chaos, like the correlation time, entropy, synchronization threshold, depend on the largest Lyapunov exponent. Therefore the coupling sensitivity is not only of theoretical interest. In this paper we study this e ect in detail. We apply an analytical approach based on the modeling of the perturbation dynamics in coupled systems with a set of linear stochastic equations (recently such an approach has been applied to coupled map lattices 10], see Sec. II H for details). For this set we get an analytical expression for the largest Lyapunov exponent, valid for arbitrary coupling and systems' parameter mismatch. This allows us to show that the logarithmic singularity disappears if the interacting systems have di erent exponents. We also obtain analytic expressions for generalized Lyapunov exponents. The theoretical predictions (Sec. II) are illustrated with numerical calculations of coupled maps and interacting high-dimensional continuous-time systems (Sec. III). Apart from the analytical treatment, we present in Sec. II D simple arguments explaining the singularity form with the help of elementary random-walk dynamics. A theoretical investigation based on modeling the uctuations of Lyapunov exponents by random noise has already been undertaken by Daido 7]. In contrast to our approach, it started from discrete-time equations and was limited to the case of coupled identical one-dimensional maps.

II. ANALYTICAL APPROACH A. Stochastic Continuous-time Model
In this section we formulate and investigate a stochastic continuous-time model for coupled chaotic systems. First, we neglect the high-dimensionality of the interacting chaotic systems and describe linear perturbations in each system with a scalar variable. In other words, we are following the perturbation corresponding to the largest Lyapunov exponent only. Second, we model the uctuations of the growth rate with a stochastic multiplicative term in the equations of motion. This approach has been succesfully applied in studies of 2

di erent statistical properties of chaos 3,11]. Summarizing, we propose the two-dimensional system of Langevin equations du1 = ( (t) + )u + "(u ? u ); (1) 1 1 1 2 1 dt du2 = ( (t) + )u + "(u ? u ) (2) 2 2 2 1 2 dt

as a continuous-time model for the linearized equations of coupled chaotic systems. Three groups of parameters describe three important ingredients of the dynamics: Lyapunov exponents of uncoupled systems are described by the constants 1;2. Fluctuations of local growth rates are modeled with the terms 1;2(t) which are random processes with zero mean values. In order to be able to apply the powerful theory of the Fokker-Planck equation 12], we assume furthermore these processes to be independent, Gaussian, and delta-correlated h ii = 0 ; h i(t) j (t0)i = 2 i2 ij (t ? t0) :
2 The parameters 1;2 describe the uctuations of local expansion rates in the chaotic 2 systems. The quantities 1;2 can be put in direct correspondence to the uncoupled chaotic systems, if one calculates the distribution of local ( nite-time) Lyapunov exponents 13]. Such a distribution has the asymptotic (for large time intervals T ) form Prob( T ) e?T ( T );

with a scaling function having its minimum at the true Lyapunov exponent . For the stochastic model (1){(2) the local Lyapunov exponents are nite-time averages of the Gaussian delta-correlated process, so that their distribution is also Gaussian: Prob( T ) e?T ( T ? )2 (2 )?2 : This means that we in fact use the parabolic approximation of the function and get the parameter 2 from this function: ?2 = 2 00 ( ):
Coupling

is described by the last terms on the r.h.s., it is proportional to the coupling constant ". For a while a symmetrical coupling is assumed, the case of asymmetrical coupling is considered in Sec. II F below. Note that in this formulation we assume the statistical properties of the underlying chaotic motion to be independent of the coupling: the parameters 1;2 and the statistical properties of the uctuations 1;2 are "-independent. This assumption is supported by the theory 14], where the invariant measure for weakly coupled systems is constructed using perturbation methods, so that the measure has no singularities in dependence on ". Thus the theory below is valid as soon as we can neglect "-dependence of the statistical properties of chaos compared to singular "-dependence of the largest Lyapounov exponent. 3

B. The Fokker-Planck Equation and the Maximal Lyapunov Exponent
Before writing the Fokker-Planck equation for the stochastic system (1){(2), we perform a transformation to new variables. First we note that for large times and positive coupling " both variables u1;2 have the same sign. Indeed, it is easy to see that the regions u1; u2 > 0 and u1; u2 < 0 are absorbing ones because for u1 = 0 we have u1 = "u2 and for u2 = 0 _ we have u2 = "u1. Thus eventually one observes the state with u1u2 > 0 independently of _ initial conditions. So the transformation

v1 = ln(u1=u2); v2 = ln(u1u2);
can be performed, leading to the equations dv1 = ? 2" sinh(v ) + 1 dt 1 dv2 = + 2" cosh(v ) + 1 dt 2
1

? 2; ? 2";

(3) (4)

1+ 2

where 1 = 1 ? 2 and 2 = 1 + 2. One can see that the dynamics of v1 is v2-independent, thus, although the noisy forcing terms 1;2 are no more statistically independent, we can write the Fokker-Planck equation for the probability density (v1; t) 12]:

@ ? ( ? ) @ + 2 2 @2 ; _ = 2" cosh(v1) + 2" sinh(v1) @v 1 2 2 @v1 @v1 1
2 2 where 2 = ( 1 + 2 )=2. The stationary solution of (5) is given by stat(v1) = C exp(lv1

"

#

(5)

?"

?2 cosh v

1 );

with the normalization constant C . Basing on the solution (6) we now calculate the largest Lyapunov exponent max (below we omit the index, denoting the largest exponent for simplicity as ), de ned by 1 = tlim 1 2 hln(u2 + u2)i : 1 2 !1 t The norm u2 + u2 can be expressed in terms of v1 and v2 as 1 2 ln(u2 + u2) = v2 + ln(2 cosh v1) : 1 2 Since one is interested in the large-time limit, the stationary distribution of v1 may be used. Because hln(2 cosh v1)i stat(v1 ) is nite and time-independent, the only contribution to the largest Lyapunov exponent comes from v2. Thus Eq. (4) gives the equation for : = 1 hv2i = "hcosh v1i + 1 ( 1 + 2 ? 2") : _ (7) 2 2 The averaging with the stationary distribution stat(v1) yields 4

where l = 12?2 2 ;

(6)

2 2 hcosh v1i = K1?jlj("= K ) + K1+jlj("= ) ; 2 ("= 2)

jlj

where Kl are modi ed Bessel Functions (Macdonald Functions) 15]. Substituting this in (7) we obtain a nal analytical formula for the largest Lyapunov exponent. We write it in a scaling form:

? 1 ( 1 + 2 ? 2") = " K1?jlj ("= 2) + K1+jlj("= 2) : 2 2 2 2K ("= 2)
jlj

(8)

This form demonstrates that the essential parameters of the problem are the coupling parameter and the Lyapunov exponents' mismatch normalized to the uctuation of the exponents: "= 2 and l = ( 1 ? 2)=(2 2), correspondingly. Simpli ed expressions can be obtained in the following limiting cases: a. Small coupling, equal Lyapunov exponents. According to (6), if the Lyapunov exponents of two interacting systems are equal, 1 = 2 = , then the parameter l vanishes and we get (cf. 10])

K ("= 2) = " K1("= 2) + ? " : 0
For small coupling "= 2 the leading term in " is singular, as it follows from the expansions of K1 and K0 15]:

?

j ln("= 2)j :

2

(9)

This formula corresponds to Daido's singular dependence of the Lyapunov exponent on the coupling parameter 4{6]. It is valid in all cases, when identical chaotic systems are coupled symmetrically, provided that the Lyapunov exponents in these systems uctuate ( 2 > 0). Moreover, even for di erent systems having however equal Lyapunov exponents (but not necessarily equal uctuations of the exponents) we get the same singularity as for identical systems. Daido arrived at a similar result in his analytical treatment of coupled one-dimensional maps, cf. Eq. (19) of Ref. 7]. b. Small coupling, di erent Lyapunov exponents. The expansion (9) remains valid for small values of mismatch jlj, if ("= 2)jlj is close to 1. For larger mismatch, when

"
the largest Lyapunov exponent is

jlj

2

1;

2jlj 1 lj) 2 2jlj ?(1 ? jlj) 2" 2 + 2 (j 1 ? 2j + 1 + 2) : (10) ?(1 + j The singularity is now of the power-law type, with the power depending on the system's mismatch. Note also that this is the correction to the largest of the Lyapunov exponents of uncoupled systems 1;2.

5

c. Large coupling. For "=

2 2

1 the expansion of (8) gives
2 4

1 ? (1 +83"l ) + 2 ( 1 + 2) : 2

(11)

C. Generalized Lyapunov Exponents
The generalized Lyapunov exponents characterize nite-time uctuations of the exponential growth rate. For our linear model (1){(2) they are de ned as 13] L(q) = tlim 1 lnh(u2 + u2)q=2i: (12) 1 2 !1 t For simplicity of presentation we assume below that the interacting systems are identical and therefore will omit the index at the parameters 2 and . It is straightforward to obtain the generalized Lyapunov exponents for integer q. For q = 1 we need equations for the mean values hu1;2i which can easily be obtained by direct averaging of the system (1){(2) using the Furutsu-Novikov relation 16,17]: d hu i = ( + 2 ? ")hu i + "hu i; (13) 1 2 dt 1 d hu i = ( + 2 ? ")hu i + "hu i: (14) 2 1 dt 2

Thus the averages hu1;2i grow exponentially and the generalized Lyapunov exponent is

L(1) = + 2: L(2) = 2 + 3 2 ? 2" +

(15)

Similarly, we can write the 3-d order system of linear equations for the moments hu2i ; hu2i ; hu1u2i and determine L(2) as the largest eigenvalue of this system: 1 2

p

4 + 4"2

:

(16)

This method works for all integer moments, but for q > 2 we have to look for roots of polynomials of order 4 and higher, so the analytical expressions are hardly available. Also, we do not have a method for calculation of the generalized exponents for non-integer indices. Having expressions for L(1) and L(2), we can nd an approximate expression for the usual Lyapunov exponent (cf. 18]). Indeed, this exponent is determined by the behavior of L(q) near q = 0: = L0(0) (this formula follows directly from (12), see also 13]). As L(q) is a convex function and L(0) = 0, knowing two points L(1) and L(2) we can approximate it with a parabola:

L(q) = q + q2;
with the parameters 6

= 2L(1) ? L(2) ; 2 ~= = + For "
2 2

= ?L(1) + L(2) : 2 +"?
4

Thus we get the approximation for the usual Lyapunov exponent
s
4

2
2

4

+ "2 :

this gives ~ 2 ? 8" + ;

what coincides with (11). We see that the parabolic approximation for the generalized exponent spectrum L(q) is valid for large couplings and small uctuations of the nite-time exponents. Another limiting case, when the form of the generalized exponent spectrum is exactly parabolic, is that of zero coupling

L(q) = q + 2q2 :
For small coupling, where the logarithmic singularity of the usual exponent (9) is essential, the parabolic approximation does not work.

D. A Qualitative Picture
Here we give qualitative arguments supporting the main singularity formula (9). Let us consider the symmetric case and small couplings (moreover, for simplicity of presentation we assume = 0). For small ", the coupling in the system (1){(2) in uences the dynamics only if the di erence between u1 and u2 is large. E.g., if u2 u1=" u1, then the coupling term in the rst equation (1) is of the same order as other terms and it contributes to the growth of the variable u1. At the same time the in uence of u1 on u2 remains small. Thus, the coupling \switches on" only rarely, but leads to e ective equalization of the variables where the smallest one is adjusted to the largest one. We illustrate this process in Fig. 1(a). To make these arguments quantitative, let us represent the same qualitative picture in the plane of logarithmic variables ln u1, ln u2 see Fig. 1(b)]. Here we have a random walk in two dimensions, and this walk is restricted to the strip j ln u1 ? ln u2j < ? ln " (the connection between the dynamics in the plane of logarithmic variables and the random walk has already been pointed out in Ref. 7]). The walk has rather strange properties of re ection at the boundaries: it springs to the diagonal ln u1 = ln u2, always in the direction of growth of ln u1 and ln u2. Due to these re ections a constant drift arises, whose velocity is easy to estimate. Indeed, for an unbiased random walk starting at the center of the strip the mean time to reach the boundary is (ln ")2= 2 19], and this is a characteristic time between re ections. Each re ection makes a contribution of order of j ln "j to the mean drift. So for the mean drift velocity we get 2=j ln "j in accordance with (9).

7

E. The Second Lyapunov Exponents
From the Fokker-Planck equation approach we have obtained the largest Lyapunov exponent. The second exponent can be found as follows. For the stochastic system (1){(2) the mean divergence of the phase volume is + + * * _ _ d ln V = @ u1 + @ u2 = + ? 2"; 1 2 dt @u1 @u2 and this quantity is just the sum of the Lyapunov exponents. Thus (17) 2 = ? 1 + 1 + 2 ? 2"; and we get for 2 the same singularity as for the largest exponent, only with another sign.

F. Asymmetrical Coupling
The more general case of asymmetrical coupling can be described by the following set of Langevin equations: du1 = ( (t) + )u + " (u ? u ) ; (18) 1 1 1 1 2 1 dt du2 = ( (t) + )u + " (u ? u ) : (19) 2 2 2 2 1 2 dt

By virtue of the scaling transformation ~ u1 = p"2 u1 ; u2 = p"1 u2 ; ~ (20) the problem can be reduced to the symmetric case: _ u u ~ u1 = ( 1 + 1 ? "1 + p"1"2)~1 + p"1"2(~2 ? u1) ; ~ p" " )~ + p" " (~ ? u ) : _ u2 = ( 2 + 2 ? "2 + 1 2 u2 ~ 1 2 u1 ~ 2 Thus, we can use the expression (8) for the largest Lyapunov exponent, leading to p" " K (p" " = 2) + K (p" " = 2) 1 1 ? 2 ( 1 + 2 ? " 1 ? "2 ) 2 = 1 2 1?jlj 12K (p" " 1+jl2j) 1 2 : (21) 2 2 1 2= jlj Here the e ective mismatch and the e ective coupling are now given by l = 21 2 ( 1 ? "1) ? ( 2 ? "2)] ; " = p"1"2 : In the case of unidirectional coupling the ansatz (20) is no more valid, but in this situation the Lyapunov exponents can easily be found directly. If, e.g., "1 = 0, then the Lyapunov exponents are 1; 2 ? "2. There is no singularity for unidirectional coupling. The results for asymmetrical coupling can straightforwardly be understood in the framework of the qualitative picture of Sec. II D. Indeed, the important quantity is the width of the strip in Fig. 1(b), and this is ?(ln "1 + ln "2). In the limiting case of unidirectional coupling the width tends to in nity, the random walk never hits the boundary, and there are no essential corrections to the uncoupled exponents. 8

G. Three Coupled Chaotic Systems
Models with many coupled identical chaotic systems have attracted large attention recently (e.g. 20,21]). As a rst step in this direction, we show here that the system of three symmetrically coupled oscillators has the same logarithmic singularity in the Lyapunov exponent as the system of two oscillators. The stochastic model has the following form du1 = ( (t) + )u + "(u + u ? 2u ) ; 1 1 2 3 1 dt du2 = ( (t) + )u + "(u + u ? 2u ) ; (22) 2 2 1 3 2 dt du3 = ( (t) + )u + "(u + u ? 2u ) ; 3 3 1 2 3 dt with uncorrelated noisy terms i. In the variables u u v1 = ln u1 ; v2 = ln u2 ; v3 = ln(u1u2u3) 3 3 the system (22) reduces to dv1 = ? ? 2" sinh(v ) ? "ev2 + "ev2?v1 ; 1 3 1 dt dv2 = ? ? 2" sinh(v ) ? "ev1 + "ev1?v2 ; (23) 2 3 2 dt dv3 = + + + 3( ? 2") + 2" cosh(v ? v ) + 2" cosh(v ) + 2" cosh(v ) : 1 2 3 1 2 1 2 dt As the equations for v1; v2 constitute a closed system, we can write the Fokker-Planck equation for the probability density (v1; v2). In the limit " ! 0 the stationary solution of this equation can be looked for in the \quasipotential" (see, e.g., 22]) form = C exp ("= 2)f (v1; v2)], which in the rst order in " gives (see 23] for details): = C exp ? " (cosh(v ? v ) + cosh(v ) + cosh(v )) : (24)
2 1 2 1 2

The largest Lyapunov exponent can, analogous to eq. (7), be represented as = ? 2" + 23" hcosh(v1 ? v2) + cosh(v1) + cosh(v2)i The averaging requires a nontrivial integral, which can be estimated in the limit " (see 23]) to give the nal formula

?2

!0

2 4 (25) 3 j ln("= 2)j : The singularity in the system of three coupled chaotic oscillators is thus of the same type as for two oscillators, cf. (9).

?

9

H. Coupled Map Lattices
In a parallel work, an approach similar to ours has been used to derive an analytic expression for the scaling of the largest Lyapunov exponent in coupled map lattices 10]. For small coupling, coupled identical maps, and positive multipliers, the authors arrive at an expression

?

where = "=(1 ? 2"). This result is similar to ours for the case of two coupled systems, Eq. (9), with the di erence that " is replaced by as the scaling parameter. The authors of Ref. 10] were also able to derive an expression for the case of multipliers with uctuating signs, 2 3 ? 2 j ln( = )j : Instead of the variance 2, the standard deviation appears in the argument of the logarithm.

j ln( = 2)j ;

2

III. NUMERICAL RESULTS
We now compare the results obtained for the system of continuous-time Langevin equations with numerical calculations for both continuous- and discrete-time deterministic systems. For the calculation of Lyapunov exponents we iterate the original as well as a set of linearized equations, and reorthonormalize the di erence vectors periodically using a modied Gram-Schmidt algorithm (see 24,25] and references therein). For the rst examples (Sec. III A-III C) we iterate a system of two di usively coupled one-dimensional maps,

x1(n + 1) = f1(x1(n)) + " f2(x2(n)) ? f1(x1(n))] ; (26) x2(n + 1) = f2(x2(n)) + " f1(x1(n)) ? f2(x2(n))] ; (27) where f1(x) and f2(x) are maps speci ed below. The linearized equations read w1(n + 1) = (1 ? ")f10 (x1(n))w1(n) + "f20 (x2(n))w2(n) ; (28) 0 (x (n))w (n) + "f 0 (x (n))w (n) : (29) w2(n + 1) = (1 ? ")f2 2 1 2 1 1 The Lyapunov exponents of the uncoupled maps are 1;2 = hln jf10 ;2ji. In the simplest 2 examples considered below the variances can be calculated as 1;2 = h(ln jf10 ;2j ? 1;2)2i=2. To have a good correspondence to the theory, we use only monotonous mappings (i.e. with a constant sign of f 0) below, so that the uctuations of the local expansion rate are the only source of irregularity of perturbations' dynamics. Another source of irregularity could be irregular changes of the sign of the derivative f 0 (as for the logistic and the tent maps). Such an irregularity is not covered by our continuous-time approach, but also leads to the logarithmic singularity of type (9), see 10].
10

A. Skewed Bernoulli Maps
We rst choose f1(x) = f2(x) = f (x) to be identical maps, where f (x) is the skewed Bernoulli map de ned as ( x f (x) = (x ? xx=x0 ? x ) if x > x0 ;; (30) )=(1 0 if x 0 0 with x 2 0; 1] and 0 < x0 < 1. For the uncoupled map, the Lyapunov exponent and the magnitude of uctuations are given by = ?x0 ln x0 ? (1 ? x0) ln(1 ? x0) (31) and 2 2 = 1 x (1 ? x ) ln x0 (32) 0 2 0 1 ? x0 ; respectively. For x0 = 1=2 we obtain the ordinary Bernoulli map. In this case, there are no uctuations of the local multipliers ( 2 = 0), and no coupling sensitivity of the Lyapunov exponents is observed. Figure 2(a) shows the di erences 1;2 ? 1;2 vs " for maps with di erent values of x0 6= 1=2. From Fig. 2(b) it can be seen that di erent curves collapse onto single lines for both exponents when plotted in the rescaled form according to (8), namely as ( 1 ? 1)= 2 vs 1=j ln("= 2)j. The resulting lines are in very good agreement with the leading term of the theoretical prediction ( 1 ? 1)= 2 = 1=j ln("= 2)j, which is also shown. No such good accordance between theory and numerical experiment is found in the case of generalized Lyapunov exponents. In Fig. 3(a,b) the results for L(1) and L(2) are shown for small values of ". Also shown are the theoretical predictions from Eqs. (15) and (16), respectively. The rough correspondence is completely lost for larger values of ", although the considerations in Sec. II C are not restricted to small ". Much better results are achieved if the derivatives fi0 in the linearized equations (28) and (29) are replaced by independent and identically distributed Gaussian stochastic variables i (i = 1; 2). Then the system of equations reads w1(n + 1) = (1 ? ")e 1(n)w1(n) + "e 2(n)w2(n) ; (33) 2 (n) 1 (n) w2(n + 1) = (1 ? ")e w2(n) + "e w1(n) ; (34) with h i (n)i = and h( i (n) ? )( j (m) ? )i = 2 2 ij nm (i; j = 1; 2). In Figs. 4(a,b) the results for L(1) and L(2), respectively, are shown together with the analytical curves. The values of and 2 were calculated by means of Eqs. (31) and (32) with the values of x0 used above for the skewed Bernoulli map. An explanation for the discrepancy between the deterministic and stochastic results is that the distribution of f 0(xi) is changed with increasing ", while the distribution of the stochastic variables i remains constant. Furthermore, f 0(x1) and f 0(x2) are not statistically independent for larger values of ". These e ects have no observable in uence in the case of usual Lyapunov exponents (Fig. 2) because of the singularity. In the case of generalized Lyapunov exponents, however, the non-singular scaling functions are much more sensitive against changes in the distribution of multipliers. 11

B. Di erent Maps
One main result of the analytical approach is that the singularity does only depend on the 2 2 average 2 = ( 1 + 2 )=2 of the uctuations of local expansion rates and on the mismatch l = ( 1 ? 2)=(2 2) of the Lyapunov exponents of the uncoupled systems. Although no singularity occurs if 2 = 0, we can expect to observe coupling sensitivity in the case of a 2 2 system with uctuations ( 1 > 0) coupled to one without uctuations ( 2 = 0), given that the mismatch l is su ciently small. In order to check this prediction, we again numerically iterate the system of Eqs. (26){ (29), now choosing two di erent maps. The rst map is again the skewed Bernoulli map f1(x) = f (x), see Eq. (30)], while the second map is de ned as

f2(x) = e 1 x (mod 1) ;

(35)

where 1 is the Lyapunov exponent of the skewed Bernoulli map f (x) see Eq. (31)]. With 2 2 this choice we have the parameters 1 > 0, 2 = 0, and l = 0. In Fig. 5 the result is compared with the previous result for two coupled identical skewed Bernoulli maps (x0 = 1=4 in either case). As expected, the logarithmic singularity is observed 2 in both cases, although the deviation j i ? ij is smaller if 2 = 0. When rescaled with the average 2, however, the curves collapse onto single lines for the rst and second Lyapunov exponents, as can be seen in Fig. 5(b).

C. Systems with Anomalous Fluctuations of Lyapunov Exponents
Daido found out that for coupled logistic maps f (x) = 4x(1 ? x) the Lyapunov exponents exhibit power-law instead of logarithmic singular behavior due to anomalous uctuations of the nite-time Lyapunov exponents 7]. Here we report a similar observation in the case of coupled strange non-chaotic attractors. Fluctuations of nite-time Lyapunov exponents is a typical feature of chaotic systems, but in some non-chaotic systems the Lyapunov exponents uctuate as well. To this class belong strange non-chaotic attractors (SNAs) that have a negative maximal Lyapunov exponent but a complex fractal structure in the phase space (see 26] and references there). The uctuations of nite-time Lyapunov exponents are present in SNAs 26], but they are much more correlated than in chaotic systems. We demonstrate below that this leads to weaker singularity in the Lyapunov exponent dependence on coupling. We studied numerically two coupled quasiperiodically forced maps having strange nonchaotic attractors, taking where ! = ( 5 ? 1)=2 is the frequency of quasiperiodic driving. The model (36) has been studied rigorously in 27,28]. The results are presented in Fig. 6. The dependence of the Lyapunov exponents on the coupling has a singularity, but this singularity contrary to (9) is a power law, with a power close to 1=2. A detailed theory needs correct account of nontrivial correlation properties of the SNA and is now in progress. 12

p

f (x) = 2:5 tanh(x)j sin(!n + )j ;

(36)

D. High-dimensional Continuous-time Systems
Daido observed the e ect of coupling sensitivity of chaos not only for coupled onedimensional maps, but also for two-dimensional discrete-time maps 6]. Here we give numerical evidence that the logarithmic singularity is also observed in in nite-dimensional and continuous-time systems. As an example we study a system of two coupled one-dimensional delay di erential equations. A delay di erential equation has an in nite number of Lyapunov exponents, and for large delays usually a nite number of exponents is positive. The system we study reads x1(t) = f (x1(t); x1(t ? )) + " x2(t) ? x1(t)] ; _ (37) x2(t) = f (x2(t); x2(t ? )) + " x1(t) ? x2(t)] ; _ (38) where f (x(t); x(t ? )) = ?x(t) + a sin x(t ? ) corresponds to the Ikeda equation, describing an optical resonator system 29]. The parameter values were chosen to be a = 3:0 and = 5:0. We integrated the coupled Ikeda equations, together with the linearized equations, using the fourth order Runge-Kutta routine. The results are presented in Figs. 7(a,b). The uncoupled Ikeda system has one positive and one zero (due to invariance to time shifts) Lyapunov exponent, all other exponents are negative. In the coupled system the two former zero exponents (the third and the fourth) are not affected by the coupling sensitivity: one exponent remains exatly zero, changes of the another one are hardly seen for small couplings. We attribute this to the fact that the zero Lyapunov exponent in an autonomous system does not uctuate. The other Lyapunov exponents (the positive one and the rst negative one), however, show the logarithmic singularity.

E. Three Coupled Chaotic Maps
For three weakly coupled chaotic systems, the leading terms in the expressions for the maximum and minimum Lyapunov exponents were shown to have the same logarithmic singularities as in the case of two coupled systems, although with a di erent factor (see Sec. II G). The singularity is observed in numerical simulations for three coupled identical skewed Bernoulli maps see Fig. 8(a)]. The factor of 4=3, however, is obviously not correct, although a rough agreement between theoretical and numerical results can be seen in Fig. 8(b). A reason for the disagreement could be the neglect of terms of order "2 when nding the stationary probability distribution, Eq. (24).

IV. CONCLUSION
In this paper we used the Langevin approach to obtain statistical properties of the Lyapunov exponents for small coupling. For the simplest system of two coupled stochastic equations it is possible to obtain an analytical expression for the largest Lyapunov exponent, for di erent values of parameters (coupling, Lyapunov exponents of uncoupled systems, uctuations of Lyapunov exponents). The logarithmic singularity, rst discovered by 13

Daido, is shown to exist even if rather di erent systems are coupled, provided their Lyapunov exponents coincide. We also give a qualitative explanation of the e ect, based on the interpretation of the perturbations' dynamics as coupled random walks. The coupling " restricts the two-dimensional walk to a strip with a width log ", with rather unusual \re ection conditions" on the strip borders. As a result the random walk (and, correspondingly, the Lyapunov exponent) gets a bias (log ")?1. It is not clear, if such an e ect can be observed in the context of other random-walk-like phenomena. We have also presented some generalizations where we do not have strict analytical results. For three coupled systems we were only able to obtain leading terms in the small coupling approximation, they are of the same inverse logarithm type as for two systems. Numerical simulations of a system with weaker stochastic properties (strange nonchaotic attractor) reveals, however, a power-law singularity, possibly due to existence of long correlations in the dynamics of perturbations. Recent results presented in Ref. 10] for coupled map lattices, including the case of uctuating multiplier signs, support the assumption that the logarithmic singularity is a very general phenomenon of coupled chaotic systems.

ACKNOWLEDGMENTS
This work was supported by the Deutsche Forschungsgemeinschaft (SFB 555) and the Volkswagenstiftung. We thank A. Politi and F. Cecconi for useful discussions.

14

REFERENCES
1] H. Fujisaka and T. Yamada, Prog. Theor. Phys. 69, 32 (1983). 2] T. Yamada and H. Fujisaka, Prog. Theor. Phys. 70, 1240 (1983). 3] A. S. Pikovsky, Z. Physik B 55, 149 (1984). 4] H. Daido, Prog. Theor. Phys. 72, 853 (1984). 5] H. Daido, Prog. Theor. Phys. Suppl. 79, 75 (1984). 6] H. Daido, Phys. Lett. A 110, 5 (1985). 7] H. Daido, Phys. Lett. A 121, 60 (1987). 8] R. Livi, A. Politi, and S. Ru o, J. Phys. A: Math,Gen. 25, 4813 (1992). 9] A. Torcini, R. Livi, A. Politi, and S. Ru o, Phys. Rev. Lett. 78, 1391 (1997). 10] F. Cecconi and A. Politi, chao-dyn/9901014 (unpublished). 11] H. Fujisaka, H. Ishii, M. Inoue, and T. Yamada, Prog. Theor. Phys. 76, 1198 (1986). 12] H. Z. Risken, The Fokker{Planck Equation (Springer, Berlin, 1989). 13] A. Crisanti, G. Paladin, and A. Vulpiani, Products of Random Matrices in Statistical Physics (Springer, Berlin, 1993). 14] L. A. Bunimovich and Y. G. Sinai, Nonlinearity 1, 491 (1988). 15] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Department of Commerce USA, Washington, D.C., 1964). 16] K. Furutsu, J. Res. NBS D667, 303 (1963). 17] Y. Novikov, JETP 47, 1919 (1964). 18] A. Becker and L. Kramer, Phys. Rev. Lett. 73, 955 (1994). 19] C. W. Gardiner, Handbook of Stochastic Methods (Springer, Berlin, 1996). 20] K. Kaneko, Physica D 41, 137 (1990). 21] A. Crisanti, M. Falcini, and A. Vulpiani, Phys. Rev. Lett. 76, 612 (1996). 22] R. Graham and T. Tel, Phys. Rev. A 31, 1109 (1985). 23] R. Zillmer, Diploma Thesis, Univ. Potsdam, 1999. 24] K. Geist, U. Parlitz, and W. Lauterborn, Prog. Theor. Phys. 83, 875 (1991). 25] A. Bjorck, Linear Algebra Appl. 197/198, 297 (1994). 26] A. Pikovsky and U. Feudel, CHAOS 5, 253 (1995). 27] C. Grebogi, E. Ott, S. Pelikan, and J. A. Yorke, Physica D 13, 261 (1984). 28] G. Keller, Fundamenta Mathematicae 151, 139 (1996). 29] K. Ikeda and K. Matsumoto, Physica D 29, 223 (1987).

15

FIGURES FIG. 1. A sketch of the perturbation dynamics in coupled systems. Curly line shows the random walk not in uenced by coupling; straight arrows demonstrate the e ect of coupling.
FIG. 2. Coupled skewed Bernoulli maps, Eq. (30). (a) The Lyapunov exponents 1 ? 1 and 2 ? 2 vs " for x0 = 1=3 (solid lines), x0 = 1=4 (dotted lines), x0 = 1=5 (dashed lines), and x0 = 1=6 (dash-dotted lines). (b) The same graphs in scaled coordinates. The long-dashed lines show the analytical results ( 1 ? 1 )= 2 = 1=j ln("= 2)j and ( 2 ? 2)= 2 = ?1=j ln("= 2)j, see Eqs. (9) and (17). FIG. 3. Generalized Lyapunov exponents for the skewed Bernoulli maps. (a): Rescaled exponent L(1) ? ]= 2 vs " for the same values of x0 as in Fig. 2. The long-dashed line shows the analytical result L(1) ? ]= 2 = 1, see Eq. (15). (b): Rescaled exponent L(2) ? 2 ]= 2 vs "= 2 for the same values of x0 p in Fig. 2. The as 2 = 3 ? 2"= 2 + 1 + 4("= 2)2, see long-dashed line shows the analytical result L(2) ? 2 ]= Eq. (16). FIG. 4. Rescaled generalized Lyapunov exponents in stochastic maps. (a) The exponent L(1) ? ]= 2 vs " for and 2 corresponding to the values of x0 used for Fig. 2. The long-dashed line shows the analytical result as in Fig. 3(a). (b)The exponent L(2) ? 2 ]= 2 vs "= 2 for and 2 corresponding to the values of x0 used for Fig. 2. The long-dashed line shows the analytical result as in Fig. 3(b). FIG. 5. Di erent maps. (a) 1 ? 1 and 2 ? 2 vs " for two coupled skewed Bernoulli maps with x0 = 1=4 (solid lines) as well as one skewed Bernoulli map with x0 = 1=4 coupled with the di erent map (35) (dotted lines). (b) ( 1 ? 1 )= 2 and ( 2 ? 2)= 2 vs 1=j ln("= 2)j for the same examples as in Fig. 5(a). The long-dashed lines show the analytical results as in Fig. 2(b). FIG. 6. The Lyapunov exponents in coupled strange nonchaotic attractors in natural coordinates (a) and in a log-log representation (b). The dashed line in (b) has slope 0:5. FIG. 7. The Lyapunov exponents in the coupled Ikeda equations, in natural (a) and scaled (b) coordinates. Open circle and open square: the splitting of the positive Lyapunov exponent; Open triangle and open rhomb: the splitting of the zero exponent; cross and star: the splitting of the closest to zero negative exponent.

16

FIG. 8. Three coupled skewed Bernoulli maps. (a) The exponents i ? i (i = 1; : : :; 3) vs " for x0 = 1=3 (solid lines), x0 = 1=4 (dotted lines), x0 = 1=5 (dashed lines), and x0 = 1=6 (dash-dotted lines). (b) The exponents ( i ? i )= 2 (i = 1; : : :; 3) vs 1=j ln("= 2)j for the same values of x0 as in (a). The long-dashed lines show the analytical results (25).

17

Zillmer et al., Fig. 1:

(b)

(a)
u2

lnu2
u2 = u1

ε

- 2ln ε

u1 = u2 ε u1

ln u1

Zillmer et al., Fig. 2:
(a)
0.04 0.3 0.2

(b)

(λi ? Λi)/σ (i = 1,2)
2

0.02

λi ? Λi (i = 1,2)

0.1 0.0 ?0.1 ?0.2

0.00

?0.02

?0.04 0.0000

0.0002

0.0004

ε

0.0006

0.0008

0.0010

?0.3 0.00

0.05

0.10

1/|ln(ε/σ )|

2

0.15

0.20

0.25

18

Zillmer et al., Fig. 3:
(a)
1.4 5.5 1.3 5.0
2

(b)

1.2

[L(2)?2Λ]/σ

[L(1)?Λ]/σ

2

4.5

1.1

4.0 1.0

0.9 0.0000

0.0002

0.0004

ε

0.0006

0.0008

0.0010

3.5 0.000

0.002

0.004

ε/σ

2

0.006

0.008

0.010

Zillmer et al., Fig. 4:
1.04

(a)

(b)
4.0 3.8

1.03
2

1.02

[L(2)?2Λ]/σ

[L(1)?Λ]/σ

2

3.6

1.01

3.4

1.00

3.2

0.99 0.0

0.1

0.2

ε

0.3

0.4

0.5

3.0 0.0

1.0

2.0

ε/σ

2

3.0

4.0

5.0

19

Zillmer et al., Fig. 5:
(a)
0.3 0.02 0.2

(b)

λi ? Λi (i = 1,2)

0.01

(i = 1,2) (λi ? Λi)/σ
0.0002 0.0004 0.0006 0.0008 0.0010
2

0.1 0.0 ?0.1 ?0.2

0.00

?0.01

?0.02 0.0000 ?0.3 0.00 0.05 0.10 0.15 0.20 0.25

ε

1/|ln(ε/σ )|

2

Zillmer et al., Fig. 6:
(a)
0.1
10
0

(b)

0.0

?0.1

10

?1

?0.2

|λ1,2? Λ1,2|
10
?2

λ1,2? Λ1,2

?0.3

?0.4

?0.5 0.00

0.01

ε

0.02

10

?3

10

?6

10

?5

10

?4

ε

10

?3

10

?2

10

?1

20

Zillmer et al., Fig. 7:
(a)
0.002 0.002

(b)

λi ? Λi (i = 1,...,6)

0.000

λi ? Λi (i = 1,...,6)
0.00002 0.00004 0.00006

0.001

0.001

0.000

?0.001

?0.001

?0.002 0.00000

ε

?0.002 0.00

0.02

0.04

0.06

0.08

0.10

0.12

1/|ln(ε)|

Zillmer et al., Fig. 8:
(a)
0.06 0.04 0.4

(b)

(i = 1,...,3) (λi ? Λi)/σ
0.0002 0.0004 0.0006 0.0008 0.0010
2

λi ? Λi (i = 1,...,3)

0.2

0.02 0.00 ?0.02 ?0.04 ?0.06 0.0000

0.0

?0.2

ε

?0.4 0.00

0.05

0.10

1/|ln(ε/σ )|

2

0.15

0.20

0.25

21


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