Heterogeneity-induced order in globally coupled chaotic systems

– Collective behavior is studied in globally coupled maps with distributed nonlinearity. It is shown that the heterogeneity enhances regularity in the collective dynamics. Low-dimensional quasi-periodic motion is often found for the mean field, even if each element shows chaotic dynamics. The mechanism of this order is due to the formation of an internal bifurcation structure, and the self-consistent dynamics between the structures and the mean field. The dynamics of globally coupled systems has been extensively and intensively studied [1][10]. Such problems naturally appear in physical and biological systems. Coupled Josephson junction array [2] and nonlinear optics with multi-mode excitation [3] give such examples, while relevance to neural and cellular networks has been discussed [4]. Among others, the study of globally coupled chaotic systems has revealed novel concepts such as clustering, chaotic itinerancy, and partial order. In particular, the study of collective dynamics has gathered much attention [8]-[13]. Even if the elements are desynchronized, some kinds of collective dynamics, ranging from low-dimensional torus to high-dimensional chaos, are observed [6], [8]. In these recent studies, elements are homogeneous. In other words, identical elements are coupled. However, in many systems elements are heterogeneous. In a Josephson junction array, each unit is not identical. In an optical system, the gain of each mode depends on its wavenumber. In a biological system, each unit such as a neuron or a cell is heterogeneous. So far the study of a coupled system with distributed parameters is restricted to synchronization of oscillators [9], [10]. Thus it is important to check how the notions constructed in globally coupled dynamical systems can be applicable to a heterogeneous case. In the present letter we demonstrate that the collective order emerges in a heterogeneous system through self-consistent dynamics between the mean-field and internal differentiation of dynamics. Here we adopt a globally coupled map with a distributed parameter: xn+1(i) = (1− )fi(xn(i)) + N N ∑ j=1 fj(xn(j)) (i = 1, 2, 3, . . . , N), c © Les Editions de Physique 418 EUROPHYSICS LETTERS Fig. 1. – a) Return map of the mean-field h. a0 = 1.9, ∆a = 0.05, = 0.11, N = 8 × 10 . b) The mean-square deviation of the mean field is plotted vs. coupling strength . a = 1.9, ∆a = 0.05. The number of elements is varied from N = 2 to 2. Collective motions are clearly seen in some parameter regimes such as ≈ 0.018, 0.038 < < 0.058, 0.083 < < 0.1, 0.105 < < 0.12, and 0.13 < < 0.15. where xn(i) is the variable of the i-th element at discrete time n, and fi(x(i)) is the internal dynamics of each element. For the dynamics we choose the logistic map fi(x) = 1 − a(i)x, where the parameter a(i) for the nonlinearity is distributed between [a0 − ∆a 2 , a0 + ∆a 2 ] as a(i) = a0 + ∆a(2i−N) 2N . We note that the essentially same behavior is found when a(i) is randomly distributed in an interval or the coupling (i) is distributed instead of a. When elements are identical with ∆a = 0, the present model reduces to a globally coupled map (GCM) studied extensively. In this case the mean field hn = 1 N ∑N j=1 f(xn(j)) is not stationary and a kind of collective dynamics is observed, when the chaotic dynamics of xn(i) are mutually desynchronized. Since the mean-field dynamics is generally complicated, the amplitude of oscillations is not easily measured as in the case for simple quasi-periodic dynamics. In this case, as a measure of the amplitude of the variation, we use the mean square deviation (MSD) of the mean-field distribution, given by 〈(δh)〉 = 〈(h− 〈h〉)〉. When elements are not identical, one might expect that collective dynamics would be destroyed and the mean field becomes stationary. On the contrary, the MSD remains finite even in the large-size limit. This implies the existence of some structure and coherence in the mean-field dynamics. In the present letter we clarify the form and the origin of such collective order in a heterogeneous system. Figure 1 a) gives an example of the return map of the mean field. Here the width of scattered points along the one-dimensional curve decreases with N . Hence the figure clearly shows that the mean-field dynamics is on a 2-dimensional torus. The power spectrum of the mean-field time series also supports that the motion is quasi-periodic. Figure 1 b) shows the MSD plotted as a function of the coupling strength with the increase of the system size. The MSD stays finite within a wide parameter region in which the power spectrum has delta peaks. Such collective behavior is rather general in our heterogeneous system and is observed more clearly than in the homogeneous GCM. With the change of a0, ∆a, or , the mean-field dynamics shows the bifurcation from torus to chaos accompanied by phase lockings. Further bifurcation proceeds to higher-dimensional chaos (while some structure is still kept). Several routes to chaos [12] are observed, including that through the doubling of torus (fig. 2). There are two cases for the collective motion, although for both cases each element oscillates chaotically without mutual synchronization. In one case (given in fig. 2) there are negative Lyapunov exponents (78 for b), for N = 500; whose t. shibata et al.: heterogeneity-induced order etc. 419