Synchronization between interacting ensembles of globally coupled chaotic maps

When two groups of globally coupled, identical chaotic maps interact, full synchronization (coherence) and different kinds of partial synchronization (clustering) can arise. In this paper, we identify the regions in parameter space that lead to coherence and clustering. The region of stability for the coherent state is shown to be independent of the number of maps in the ensembles. The borders of stability for the coherence region are obtained in terms of the transverse Lyapunov exponents and the nonlinearity parameter of the individual map. It is shown that for values of the nonlinearity parameter that correspond to periodic windows, the coherent state is stable in the unit square of the coupling parameter plane. For the simplest non-trivial case of four maps, we analyse the stability properties of the two-cluster states in detail and show cluster stability diagrams for particular values of the nonlinearity parameter of the logistic map. In the general case of 2 N maps, we determine the invariant cluster manifolds in phase space and identify the corresponding cluster states. For each cluster state, we study the character of the in-cluster dynamics, which is found to be regular or chaotic.