Delays, Connection Topology, and Synchronization of Coupled Chaotic Maps

Abstract

We consider networks of coupled maps where the connections between units involve time delays. We show that, similar to the undelayed case, the synchronization of the network depends on the connection topology, characterized by the spectrum of the graph Laplacian. Consequently, scalefree and random networks are capable of synchronizing despite the delayed flow of information, whereas regular networks with nearest-neighbor connections and their small-world variants generally exhibit poor synchronization. On the other hand, connection delays can actually be conducive to synchronization, so that it is possible for the delayed system to synchronize where the undelayed system does not. Furthermore, the delays determine the synchronized dynamics, leading to the emergence of a wide range of new collective behavior which the individual units are incapable of producing in isolation. (http://link.aps.org/abstract/PRL/v92/e144101) Recent years have witnessed a growing interest in the dynamics of interacting units. Particularly, a large number of studies have been devoted to synchronization in a variety of systems (see [1] and the references therein), including the coupled map lattices introduced by Kaneko [2]. Usually, such systems have been investigated under the assumption of a certain regularity in the connection topology, where units are coupled to their nearest neighbors or to all other units. Lately, more general networks with random, small-world, scale-free, and hierarchical architectures have been emphasized as appropriate models of interaction [3,4,5,6,7]. On the other hand, realistic modeling of many large networks with non-local interaction inevitably requires connection delays to be taken into account, since they naturally arise as a consequence of finite information transmission and processing speeds among the units. Some numerical studies have regarded synchronization under delays for special cases such as globally coupled logistic maps [8] or carefully chosen delays [9]. In this Letter we consider synchronization of coupled chaotic maps for general network architectures and connection delays. Because of the presence of the delays, the constituent units are unaware of the present state of the others; so it is not evident a priori that such a collection of chaotic units can operate in unison, i. e. synchronize. Based on analytical calculations, we show that this is indeed possible, and in fact may be facilitated by the presence of delays. Moreover, while the connection topology is important for synchronization, the delays have a crucial role in determining the resulting collective dynamics. As a result, the synchronized system can exhibit a plethora of new behavior in the presence of delays. We illustrate the results by numerical simulation of large networks of logistic maps.

Maximum performance at minimum cost in network synchronization

“…In addition, for a large class of systems in which the dynamics of each oscillator is chaotic, it has also been found [18,28,37,41] that…”

Network synchronization: Spectral versus statistical properties

“…2 Here we neglect any coupling delays in the network. The synchronous solutions can be markedly different when delays are introduced; see [22,23].…”

Chaotic maps coupled with random delays: Connectivity, topology, and network propensity for synchronization

“…F. Atay et al (Ata04) found, in the case of fixed (constants) delays that scale-free and random networks exhibit better synchronization properties than regular networks. More recently, A. Motter et al (Mot05) identified that the synchronization of complex networks follows a diffusive mechanism where the mean connectivity plays a key role.…”