Ju n 20 04 SYNCHRONIZATION REGIMES IN SYSTEMS WITH MULTIPLE TIME - DELAYS

We present the first report on chaos synchronization between two unidirectionally linearly and non-linearly coupled systems with multiple time-delays and using the Razumikhin-Lyapunov approach find the existence and stability conditions for different synchronization regimes. The approach is tested on the famous nonlinear models-Ikeda, Mackey-Glass and Lang-Kobayashi systems.There is continued growth in the field of chaos control [1] and ever-increasing appreciation of its appications among researchers. Application of chaos control can be found in secure communication, optimization of nonlinear system performance, modeling brain activity and pattern recognition phenomena [2], species population control [3],etc. Recently delay differential equations (DDEs) have attracted a lot of attention in the field of nonlinear dynamics. DDEs are used to model dynamical systems in many scientific and enginering areas,e.g.optics, biology, climatology, economy, cryptosystems based on synchronized hyperchaos, networks [4-5], etc. In comparison with a single time-delay DDEs with multiple time-delays are more relaistic models in the interacting complex systems. These delays result naturally from the finite propagation velocity of information, from the latency of feedback loops, from the finite switching times between different states of the system. Additional time-delays could be useful e.g. to stabilize nonlinear system's output [6].The application possibilities based on chaos require proper control of complexity. To the best our knowledge chaos synchronization between the systems with multiple time-delays has not been investigated yet. Having in mind enormous application implications of chaos synchronization e.g. in secure communication, performance optimization in nonlinear systems, stabilization problems, etc. investigation of synchronization regimes in the multiple time-delay systems is of paramount importance. In this Letter we present the first report on chaos synchronization between two unidirec-tionally linearly and nonlinearly coupled chaotic systems with mutiple time-delays and find the existence and stability conditions for different synchronization regimes. We test the approach on the paradigm Ikeda, Mackey-Glass and Lang-Kobayashi models [4]. We hope this research will pave the way for the intensive experimental investigations of chaos synchronization in the multple time-delays systems. General approach.-Consider synchronization between the double-feedback systems of general