Random Coupling of Chaotic Maps leads to Spatiotemporal Synchronisation

Abstract

We investigate the spatiotemporal dynamics of a network of coupled chaotic maps, with varying degrees of randomness in coupling connections. While strictly nearest neighbour coupling never allows spatiotemporal synchronization in our system, randomly rewiring some of those connections stabilises 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 synchronised fixed point in coupling parameter space. Further, the coupling $\epsilon_{bifr}$ at which the onset of spatiotemporal synchronisation occurs, scales with the fraction of rewired sites p as a power law, for 0.1 < p < 1. We also show that the regularising 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 $\epsilon_{bifr}$ are obtained.