Coupled maps on trees

Abstract

We study coupled maps on a Cayley tree, with local (nearest-neighbor) interactions, and with a variety of boundary conditions. The homogeneous state (where every lattice site has the same value) and the node-synchronized state (where sites of a given generation have the same value) are both shown to occur for particular values of the parameters and coupling constants. We study the stability of these states and their domains of attraction. As the number of sites that become synchronized is much higher compared to that on a regular lattice, control is easier to effect. A general procedure is given to deduce the eigenvalue spectrum for these states. Perturbations of the synchronized state lead to different spatio-temporal structures. We find that a mean-field like treatment is valid on this (effectively infinite dimensional) lattice.PACS number(s): 05.45. +b, 47.20. Ky

Synchronization of chaotic systems

“…In 1995 and 1996, Gade et al 41 and Gade, 42 respectively, showed that synchronization in certain types of networks could have a general form for the analysis of the system's stability. In the 1995 paper, they showed that coupled maps on tree-structured networks would have better synchronization robustness than the same chaotic maps on a regular (e.g., nearest neighbor coupled) lattice.…”

Delays, Connection Topology, and Synchronization of Coupled Chaotic Maps

“…A periodic attractor of period 3 exists near ε = 0.945, which transforms to intermittent behavior at ε = 0.95, shown in Figure 5. All these dynamics are exhibited by the coupled system (1), since Figure 1 implies that for these parameter values the system is synchronized (provided it has an appropriate connection topology), so the collective behavior is described by (3). Clearly, delays can make the dynamics of the synchronized system quite sensitive to the coupling strength, a feature that is absent in undelayed networks.…”

“…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.…”

Global Analysis of Synchronization in Coupled Maps

“…and different types of synchronizations have arisen such as phase synchronization or complete synchronization, etc. Coupled maps [11] have been used to a great extent to understand synchronization in particular [12,13,14,15,16,17,18,19,20,21,27] and complexity in dynamical systems [22,23,24] in general. The paradigm here consists of identical individual maps, typically iterates of a functional equation like the logistic one that produce chaotic dynamics.…”