Synchronization of Four Identical Nonlinear Systems with Time-delay

This paper considers the synchronization problem for four identical nonlinear systems coupled with time-delay. We have already studied the synchronization problem for bidirectional two coupled systems with delays and derived sufficient conditions to synchronize the systems. In this paper, these approaches are extended for four identical chaotic systems unidirectionally or bidirectionally coupled using state or output feedback with time-delays. Firstly, we show, using the small-gain theorem, that trajectories of coupled strictly semi-passive systems converge to a bounded region. Then we derive sufficient conditions for synchronization of coupled systems. The derived conditions are based on the delay-dependent Lyapunov-Krasovskii approach, and the criteria are obtained in the form of linear matrix inequalities (LMIs). The effectiveness of the derived conditions is illustrated by numerical examples.