Disconnected Synchronized Regions of Complex Dynamical Networks

This technical note addresses the synchronized region problem, which is converted to a more convenient matrix stability problem, for complex dynamical networks. For any natural number $n$ , the existence of a network with $n$ disconnected synchronized regions is theoretically proved and numerically demonstrated. This shows the intrinsic complexity of the network synchronization problem. Convexity characteristic of stability for relevant matrix pencils is further discussed. A smooth Chua's circuit network is finally discussed as an example for illustration.