Research Of Cluster Synchronization Of Discontinuous Dynamical Systems On Complex Networks

Synchronization is the term to address an identical state occupied by two or more dynamically coupled subsystems after a transient process. This phenomenon gener-ally appears in coupled dynamical systems, such as biological, circuit, physical, chemical and even social systems. It is induced by the coupling among the node dynamics in a net-work. As the progress on the investigation of it, many different types of synchronization have been observed. They are the complete synchronization, phase synchronization, lag synchronization and partial synchronization and so on. The partial synchronization has attracted much attention of researchers since its valuable applications in the brain science, technology, ecology and social science. The great majority of the previous researches fo-cused mainly on networks consisting of typical dynamical systems which are continuous or smooth every where. Besides those, there is the other type of systems, i.e., discontin-uous systems, in which the dynamical variables often under go abrupt or discontinuous changes in some regions of the phase space. The examples include nerve cells, electronic circuits, relaxation oscillators, and impact oscillators. Therefore, it has both the academic and the practical values to investigate the dynamics of smooth systems and the synchro-nized clusters in the coupled systems of discontinuous subsystems.The main achievement of the current dissertation are as follows:1. The synchronous patterns are studied in a coupled piece-wise linear maps. A new type of dynamical pattern, the cyclic synchronous patterns addressed by us, is observed, and its dynamical features and the mechanism that governs the producing of this pattern are analyzed. The characteristics of the traveling-wave is also studied. It is shown that the coupling strength dependence of the traveling-wave velocities shows a phase lock be-havior, which provides a new evidence of multiple devil’s staircase. Furthermore, the influence of the external noise on the collective dynamics of the coupled discontinuous systems and the controllability of cyclic pattern are checked, and. The potential applica-tion of the cyclic synchronization is discussed, which gives rise to a scheme of the secure communication and encrypt.2. The synchronous patterns in complex networks are inquired. A cyclic synchronous pattern is also observed and the mechanism is explained. The effects of the degree of nodes, the shortest path and the coupling strength on the propagation behavior of the trav-eling wave are discussed.3. The synchronization phenomenon in globally coupled discontinuous systems is studied. The coupling strength dependence of the synchronization order parameter shows that there is a transition process before synchronization and a none-complete synchronous window is observed in the synchronized range. The coexistence of the periodic cluster synchronization and the chaotic cluster synchronization is observed in the transition pro-cess. The former represents that there are two clusters with their states of nodes synchro-nized to two different periodic orbits, respectively. While the latter shows that there are two clusters with the states of the nodes synchronized to two different chaotic orbits. In the none-complete synchronous window, the coexistence of the cluster synchronous peri-odic state and synchronized chaotic state appears. The mechanisms of these co-existences are understood and the influence of the noise on it is also discussed.4. The critical coupling strength of the complete synchronization in the coupled discontinuous systems is studied, which reveals the limitation on the conventional method for calculating the critical coupling strength. Based upon the mechanism of the periodic cluster synchronous, an analytical method for calculating the critical coupling strength is proposed and the sufficient conditions for the complete synchronization are found.