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Random Coupling of Chaotic Maps Leads to Spatiotemporal Synchronization

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Random Coupling of Chaotic Maps Leads to Spatiotemporal Synchronization PHYSICAL REVIEW E 66, 016209 ͑2002͒ Random coupling of chaotic maps leads to spatiotemporal synchronization Sudeshna Sinha* Institute of Mathematical Sciences, Taramani, Chennai 600 113, India ͑Received 8 March 2002; published 18 July 2002͒ We investigate the spatiotemporal dynamics of a network of coupled chaotic maps, with varying degrees of randomness in coupling connections. While strictly nearest neighbor coupling never allows spatiotemporal synchronization in our system, randomly rewiring some of those connections stabilizes entire networks at x*, where x* is the strongly unstable fixed point solution of the local chaotic map. In fact, the smallest degree of randomness in spatial connections opens up a window of stability for the synchronized fixed point in coupling ⑀ parameter space. Further, the coupling bifr at which the onset of spatiotemporal synchronization occurs, scales with the fraction of rewired sites p as a power law, for 0.1ϽpϽ1. We also show that the regularizing effect of random connections can be understood from stability analysis of the probabilistic evolution equation for the ⑀ system, and approximate analytical expressions for the range and bifr are obtained. DOI: 10.1103/PhysRevE.66.016209 PACS number͑s͒: 05.45.Ra, 05.45.Xt I. INTRODUCTION is chosen to be the fully chaotic logistic map, f (x)ϭ4x(1 Ϫx). This map has widespread relevance as a prototype of The coupled map lattice ͑CML͒ was introduced as a low dimensional chaos. simple model capturing the essential features of nonlinear Now we will consider the above system with its coupling dynamics of extended systems ͓1͔. Over the past decade re- connections rewired randomly in varying degrees, and try to search centered around CML has yielded suggestive concep- determine what dynamical properties are significantly af- tual models of spatiotemporal phenomena, in fields ranging fected by the way connections are made between elements. from biology to engineering. In particular, this class of sys- In our study, at every update we will connect a fraction p of tems is of considerable interest in modeling phenomena as randomly chosen sites in the lattice, to two other random diverse as Josephson junction arrays, multimode lasers, vor- sites, instead of their nearest neighbors as in Eq. ͑1͒. That is, tex dynamics, and even evolutionary biology. The ubiquity we will replace a fraction p of nearest neighbor links by of distributed complex systems has made the CML a focus of random connections. The case of pϭ0 corresponds to the sustained research interest. usual nearest neighbor interaction, while pϭ1 corresponds A very well studied coupling form in CMLs is nearest to completely random coupling ͓2–7͔. neighbor coupling. While this regular network is the chosen This scenario is much like small world networks at low p, topology of innumerable studies, there are strong reasons to namely, pϳ0.01. Note, however, that we explore the full revisit this fundamental issue in the light of the fact that range of p here. In our work 0рpр1. So the study is inclu- some degree of randomness in spatial coupling can be closer sive of, but not confined to, small world networks. to physical reality than strict nearest neighbor scenarios. In fact, many systems of biological, technological, and physical significance are better described by randomizing some frac- II. NUMERICAL RESULTS tion of the regular links ͓2–7͔. So here we will study the We will now present numerical evidence that random re- spatiotemporal dynamics of CMLs with some of its coupling wiring has a pronounced effect on spatiotemporal synchroni- connections rewired randomly, i.e., an extended system com- zation. The numerical results here have been obtained by prized of a collection of elemental dynamical units with sampling a large set of random initial conditions (ϳ104), varying degrees of randomness in its spatial connections. and with lattice sizes ranging from 10 to 1000. Specifically, we consider a one-dimensional ring of Figures 1 and 2 display the state of the network, xn(i), coupled logistic maps. The sites are denoted by integers i iϭ1,..., N, with respect to coupling strength ⑀, for the limit- ϭ1,...,N, where N is the linear size of the lattice. On each ing cases of nearest neighbor interactions ͑i.e., pϭ0͒ and site is defined a continuous state variable denoted by xn(i), completely random coupling ͑i.e., pϭ1͒. It is clearly seen which corresponds to the physical variable of interest. The that the standard nearest neighbor coupling does not yield a evolution of this lattice, under standard nearest neighbor in- spatiotemporal fixed point anywhere in the entire coupling teractions, in discrete time n is given by range 0р⑀р1 ͓8͔. Now the effect of introducing some random connections, ⑀ i.e., pϾ0, is to create windows in parameter space where a x ϩ ͑i͒ϭ͑1Ϫ⑀͒f ͓x ͑i͔͒ϩ ͕x ͑iϩ1͒ϩx ͑iϪ1͖͒. n 1 n 2 n n spatiotemporal fixed point state gains stability, i.e., where ͑ ͒ ϭ ϭ 1 one finds all lattice sites synchronized at xn(i) x* 3/4, for all sites i and at all times n. Note that x*ϭ f (x*) is the fixed The strength of coupling is given by ⑀. The local on-site map point solution of the individual chaotic maps, and is strongly unstable in the local chaotic map. We then have for all p Ͼ0, a stable region of synchronized fixed points in the pa- ⑀ р⑀р ⑀ *Email address: [email protected] rameter interval, bifr 1.0. The value of bifr , where the 1063-651X/2002/66͑1͒/016209͑6͒/$20.0066 016209-1 ©2002 The American Physical Society SUDESHNA SINHA PHYSICAL REVIEW E 66, 016209 ͑2002͒ FIG. 1. Bifurcation diagram showing values of xn(i) with re- spect to coupling strength ⑀, for coupled logistic maps with strictly regular nearest neighbor connections. Here the linear size of the ϭ ϭ lattice is N 100 and in the figure we plot xn(i)(i 1,...,100) over nϭ1,...,5 iterations ͑after a transience time of 1000͒ for five differ- ent initial conditions. FIG. 3. The stable range R with respect to the fraction of ran- domly rewired sites p (0.001рpр1). The solid line displays the analytical result of Eq. ͑7͒, and the different points are obtained spatiotemporally invariant state onsets, is dependent on p.It from numerical simulations over several different initial conditions, ⑀ is evident from Fig. 2 that bifr for completely random cou- for four different lattice sizes, namely, Nϭ10, 50, 100, and 500. pling pϭ1 is around 0.62. The dotted line shows Rϭp, and it is clear that for a large range of The relationship between the fraction of rewired connec- p the approximation holds. Rϭ Ϫ⑀ tions p and the range, (1 bifr), within which spa- tiotemporal homogeniety is obtained, is displayed in Fig. 3. It is clearly evident that unlike nearest neighbor coupling, random coupling leads to large parameter regimes of regular homogeneous behavior, with all lattice sites synchronized exactly at x(i)ϭx*ϭ0.75. Furthermore, the synchronized spatiotemporal fixed point gains stability over some finite parameter range under any finite p, i.e., whenever pϾ0, however small, we have RϾ0. In that sense strictly nearest neighbor coupling is singular as it does not support spa- tiotemporal synchronization anywhere in coupling parameter ⑀ ͑ FIG. 4. The bifr i.e., the value of coupling at which the onset of spatiotemporal synchronization occurs͒ with respect to fraction of randomly rewired sites p (0.001рpр1). The solid line displays the ͑ ͒ FIG. 2. Bifurcation diagram showing values of xn(i) with re- analytical result of Eq. 6 , and the points are obtained from nu- spect to coupling strength ⑀, for coupled logistic maps with com- merical simulations over several different initial conditions, for four pletely random connections. Here the linear size of the lattice is different lattice sizes, namely, Nϭ10, 50, 100, and 500. The inset ϭ ϭ Ͻ р N 100 and in the figure we plot xn(i)(i 1,...,100) over n box shows a blow up of 0.1 p 1. Here the numerically obtained ϭ ͑ ͒ ⑀ 1,...,5 iterations after a transience time of 1000 for five different bifr deviates from the mean field results. The dashed line is the best initial conditions. fit straight line for the numerically obtained points in that region. 016209-2 RANDOM COUPLING OF CHAOTIC MAPS LEADS TO . PHYSICAL REVIEW E 66, 016209 ͑2002͒ space, whereas any degree of randomness in spatial coupling ⑀ h ϩ ͑ j͒ϭ͑1Ϫ⑀͒f Ј͑x*͒h ͑ j͒ϩ͑1Ϫp͒ ͕h ͑ jϩ1͒ connections opens up a synchronized fixed point window. n 1 n 2 n Thus random connections yield spatiotemporal homogeneity here, while completely regular connections never do. ⑀ ϩ ͑ Ϫ ͖͒ϩ ͕ ͑␰͒ϩ ͑␩͖͒ ⑀ hn j 1 p hn hn The relationship between bifr , the point of onset of the 2 spatiotemporal fixed phase, and p is shown in Fig. 4. Note Ͻ ⑀ ⑀ that for p 0.1 random rewiring does not affect bifr much. Ϸ͑1Ϫ⑀͒f Ј͑x*͒h ͑ j͒ϩ͑1Ϫp͒ ͕h ͑ jϩ1͒ ϳ ⑀ n n Only after p 0.1 does bifr fall appreciably. Further, it is 2 clearly evident that for 0.1Ͻpр1 the lower end of the sta- ϩh ͑ jϪ1͖͒, ͑3͒ bility range falls with increasing p as a well defined power n ⑀ law. Note that lattice size has very little effect on bifr , and ⑀ as to a first approximation one can consider the sum over the the numerically obtained bifr for ensembles of initial random fluctuations of the random neighbors to be zero. This ap- ϭ initial conditions over a range of lattice sizes N 10, 50, 100, proximation is clearly more valid for small p. and 500 fall quite indistinguishably around each other.
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