Complete synchronization and delayed synchronization in couplings. (English) Zbl 1345.34098
Summary: 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 regarding their ability to synchronize either completely or with delay: Symmetric Linear Coupled System, Commanded Linear Coupled System, Commanded Coupled System with delay and symmetric coupled system 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 synchronization. We also obtain analytically the Window of chaotic delayed synchronization for the only considered coupling 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.
MSC:
| 34D06 | Synchronization of solutions to ordinary differential equations |
| 37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |
| 37M05 | Simulation of dynamical systems |
| 37N35 | Dynamical systems in control |
| 39A33 | Chaotic behavior of solutions of difference equations |
| 39A30 | Stability theory for difference equations |
| 39A23 | Periodic solutions of difference equations |
Keywords:
couplings of chaotic dynamical systems; complete synchronization; delayed synchronization; windows of synchronizationReferences:
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