Stability of synchronized and clustered states in a system of coupled piecewise-linear maps

Abstract

Parameter regions for different types of stability of synchronized and clustered states are obtained for two interacting ensembles of globally coupled one-dimensional piecewise-linear maps. We analyze the strong (asymptotic) and weak (Milnor) stability of the synchronized state, as well as its instability. We establish that the stability and instability regions in the phase space depend only on parameters of the individual skew tent map and do not depend on the ensemble size. In the simplest nontrivial case of four coupled chaotic maps, we obtain stability regions for coherent and two-cluster states. The regions appear to be large enough to provide an efficient control of coherent and clustered chaotic regimes. The transition from desynchronization to synchronization is identified to be qualitatively different in smooth and piecewise-linear models.