Disconnected Synchronized Regions of Complex Dynamical Networks

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> 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 <formula formulatype="inline"><tex Notation="TeX">$n$</tex> </formula>, the existence of a network with <formula formulatype="inline"> <tex Notation="TeX">$n$</tex></formula> 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. </para>