Synchronization analysis of polytopic complex dynamical network

Abstract

Complex dynamical networks are ubiquitous in our real world, ranging from biological, physical, to social networks. Over the past decade, much of the interesting dynamical behaviour of complex dynamical networks, such as synchronization, spatiotemporal chaos, auto-waves and spiral waves, has recently attracted increasing attention from researchers in different areas. Among these, synchronization has been a hot research topic in recent years. Synchronization behaviour of networks is a universal phenomenon in nature; while, synchronization techniques have been widely applied in our daily life. Hence, there is a great demand to study synchronization behaviour of complex dynamical networks. There are some common phenomena in most real-world networks including the diversity of complex networks, structural uncertainty and time-varying links. Research on the corresponding concerned issues of synchronization analysis of complex dynamical networks will be presented in this thesis. The research problems are as follows: (a) Is it possible to establish synchronization criteria of a group of complex networks? (b) How does one establish synchronization criteria of complex networks with structural uncertainty? (c) How does one propose synchronization criteria of time-varying complex networks? Problem (a) arises from the fact that in practice people always have to investigate the synchronization of many different networks. The existing results can check the synchronization behaviour for only one complex network, which motivates the research of Problem (a). In this thesis, the concept of the polytopic complex network family is first defined. The models of inner and outer network families are constructed. The coupling delay, a common phenomenon of real life networks, is also considered in these models. The delay independent and delay dependent synchronization criteria are derived for inner and outer network families, respectively. Compared with the existing results, the proposed synchronization criteria in this thesis are satisfied by a set of complex networks within the same family, rather than being applied for only single complex network. Since intrinsic physical disturbances exist in real-life networks, the structural uncertainty of complex networks is unavoidable. Thus, Problem (b) arises, which few results have addressed. In this thesis, some sufficient conditions to guarantee the synchronization of complex networks with structural uncertainty will be derived. The uncertainty is considered in inner and outer coupling matrices of complex networks, which represent the network structure. Both polytopic and norm-bounded representations of uncertainty are discussed in detail. It is widely known that many real networks have time-varying structure. Hence, Problem (c) is investigated. In this thesis, the time-varying polytopic complex network is constructed. Based on the parameter-dependent Lyapunov function, the global synchronization criterion is proposed. In our result, the condition on the negative definition of some time-varying matrices has been removed, thus providing a more convenient way to verify the condition when compared with existing results.