Complete synchronization and delayed synchronization in couplings

Abstract

We consider a general coupling of two identical chaotic dynamical systems, and we obtain the conditions for synchronization. We consider two types of synchronization: complete synchronization and delayed synchronization. Then, we consider four different couplings having different behaviors regard- ing their ability to synchronize either completely or with delay: Symmetric Linear Coupled System, Com- manded Linear Coupled System, Commanded Cou- pled System with delay and symmetric coupled sys- tem with delay. The values of the coupling strength for which a coupling synchronizes define its Window of synchronization. We obtain analytically the Windows of complete synchronization, and we apply it for the considered couplings that admit complete synchroniza- tion. We also obtain analytically the Window of chaotic delayed synchronization for the only considered cou- pling that admits a chaotic delayed synchronization, the commanded coupled system with delay. At last, we use four different free chaotic dynamics (based in tent map, logistic map, three-piecewise linear map and cubic-like map) in order to observe numerically the analytically predicted windows.