Polynomial Lyapunov Functions for Synchronization of Nonlinearly Coupled Complex Networks

In this article, we search for polynomial Lyapunov functions beyond the quadratic form to investigate the synchronization problems of nonlinearly coupled complex networks. First, with a relaxed assumption than the quadratic condition, a synchronization criterion is established for nonlinearly coupled networks with asymmetric coupling matrices. Compared with the existing synchronization criteria, our results are less conservative and have a wider application. Second, the synchronization problem for polynomial networks is characterized as the sum-of-squares (SOS) optimization one. In this way, polynomial Lyapunov functions can be obtained efficiently with SOS programming tools. Furthermore, it is shown that the local synchronization of certain nonpolynomial networks can also be analyzed by using the SOS optimization method through the Taylor series expansion. Finally, three numerical examples are presented to verify the effectiveness and less conservatism of our analytical results.

I. Introduction

In the past two decades, the research of complex networks as an interdisciplinary discipline has attracted great attention in the fields of physics, control science, social science, biology, economics, etc. [1]–[10]. Synchronization is a kind of common collective phenomenon in nature, such as the chorus of frogs in the summer night, the coordinated fluctuations of the neural networks, and the consistency of the movement frequency. Hence, it is extremely interesting to research synchronization analysis in complex dynamical networks [11]–[19]. Moreover, synchronization has a quite wide range of applications for various subjects, including social sciences [20], [21] and control science and engineering [22]–[25] in the last few years.