Synchronization Of Chaotic System Under Different Dynamic Coupling Mode

In this paper, dynamics of the chaotic systems have been studied under different coupling modes, such as phase synchronization, complete synchronization, anti-phase synchronization, amplitude death, time-space synchronization and chaotic synchronization on complex networks. On one hand, we explore rich dynamics of the chaotic systems under different coupling modes; on the other hand, we try to reveal the underlying reasons of these behavious.This paper is arranged as follows:In the first chapter, the background and theory of the chaotic dynamics system are introduced, including some typical chaotic oscillators, analysis method for chaos, various phenomena of chaos synchronization and the acknowledge of complex networks. This knowledge is the preparation for the following discussions.In the second chapter, phase synchronization of one-dimensional coupled map lattices with periodic boundary conditions are explored numerically. Fixing self-oscillation frequency, different order of oscillators will lead to different critical coupling strength Kc needed by global phase synchronization. If the oscillations are arranged according to self-frequencis from small to large on the ring, the Kc needed is the larger than that of all the other orders. If the seriation is broken and the oscillators with small self-frequency is removed to the interspace of the oscillators with large self-frequency, or say the oscillators with large and small self-frequency are distributed equably on the ring, the Kc needed will be decreased largely. Moreover, using analytic method, we deduce the equation solving the lower limit of Kc for all the orders.In the third chapter, we develop a model of chaotic systemswith two couplings to investigate the dynamics of attractive and repulsive couplings, depending on whether the couplings are both negative, both positive or opposite in sign. Attractive and repulsive couplings widely exist in reality systems and the complex interactions between them have great influence on the systems. For example, cardiac excitation-contraction coupling can induce reentrant arrhythmias which may be lethal for their rapid rate. The attractive and repulsive couplings give rise to rich phenomena, such as complete synchronization, periodic anti-phase synchronization, chaotic anti-phase synchronization and amplitude death etc. The relationship among various dynamics and possible transitions to AD are illustrated. When the system is in the maximally stable AD, we observe the transient behavior of in-phase (high frequency) and out-of-phase (low frequency) motions. The mechanism behind the phenomenon is given.In the fourth chapter we investigate the dynamics of coupled chaotic (logistic) maps in which the connections are rewired randomly with varying rewiring probability and rewiring time period. Rewiring the network after T dynamical updates of the local maps contributes to the stability of synchronized cycles in a weak coupling range. We analyse the reason, which is because unstable kink-patterns emerge and need some time steps to vanish. Further, we give the synchronized basin size B and find that in the range of small values of T, even T has relatively large B. Moreover, we also compute the average time of synchronization needed with respect to different T and find T=2 is the most efficient rewiring time period.In the fifth chapter, the synchronization between two complex networks is explored numerically. Since the small-world and scale-free properties proposed by Watts and Strogatz in Nature and Barabasi and Albert in Science respectively, complex networks have been widely investigated. By adding an intermediate connection between two complex networks, we propose a near optimal connection strategy to improve the synchronizability of the whole network after connected. Through numerical simulations on scale-free and small-world and random networks of large size, the proposed strategy is proved to be near the optimal connection and much better than random connection. Moreover, we compare the influence of coupling strength intra-network and inter-network on enhancing synchronizability. A better effect on synchronizability is obtained by enhancing coupling strength inter-network than by enhancing coupling strength intra-network. We also find the synchronizability will not infinitely grow but has an upper limit, no matter how large the inter-network coupling strength is increased the reason of which is analyzed.In the sixth chapter, the summary of the whole articles are present.