Efficient synchronization with chaotic quadratic maps

Abstract

We present a systematic way to design unidirectional and bidirectional coupling schemes for synchronizing arbitrary pairs of (identical or different) discrete dynamical systems. If the coupled chaotic systems are very similar or identical, using the singular value decomposition, it is possible to suppress the exponential divergence of the dynamics of the synchronization error, and exploit the existing contraction properties of the given systems. When non-identical systems are coupled, in order to achieve synchronization it is necessary to employ some other techniques from linear algebra, stability theory and control. We use two methods to study the stability of synchronous state: the linear stability and the Lyapunov functional analysis. In order to illustrate these methods, we use a system of two coupled chaotic quadratic maps. The map obtained by coupling exhibits a richer dynamics that the single quadratic map, but is still possible to study its behaviour.