Disconnected synchronized regions of complex dynamical networks

TL;DR

This paper investigates the structure of synchronized regions in complex dynamical networks, demonstrating the existence of networks with multiple disconnected synchronized regions and discussing stability properties with practical circuit examples.

Contribution

It proves the existence of networks with any number of disconnected synchronized regions and explores stability characteristics in matrix pencils.

Findings

Existence of networks with n disconnected synchronized regions for any natural number n

Convexity properties of matrix pencil stability are analyzed

Illustrative examples include smooth and generalized smooth Chua's circuit networks

Abstract

This paper addresses the synchronized region problem, which is reduced to a matrix stability problem, for complex dynamical networks. For any natural number $n$, the existence of a network which has $n$ disconnected synchronized regions is theoretically demonstrated. This shows the complexity in network synchronization. Convexity characteristic of stability for matrix pencils is further discussed. Smooth and generalized smooth Chua's circuit networks are finally discussed as examples for illustration.