Stability regions for synchronized τ-periodic orbits of coupled maps with coupling delay τ

Motivated by the chaos suppression methods based on stabilizing an unstable periodic orbit, westudy the stability of synchronized periodic orbits of coupled map systems when the period of theorbit is the same as the delay in the information transmission between coupled units. We show thatthe stability region of a synchronized periodic orbit is determined by the Floquet multiplier of theperiodic orbit for the uncoupled map, the coupling constant, the smallest and the largest Laplacianeigenvalue of the adjacency matrix. We prove that the stabilization of an unstable τ-periodic orbitvia coupling with delay τ is possible only when the Floquet multiplier of the orbit is negative andthe connection structure is not bipartite. For a given coupling structure, it is possible to find thevalues of the coupling strength that stabilizes unstable periodic orbits. The most suitableconnection topology for stabilization is found to be the all-to-all coupling. On the other hand, anegative coupling constant may lead to destabilization of τ-periodic orbits that are stable for theuncoupled map. We provide examples of coupled logistic maps demonstrating the stabilization anddestabilization of synchronized τ-periodic orbits as well as chaos suppression via stabilization of asynchronized τ-periodic orbit.