Synchronization And Wave Patterns In A Population Of Chaotic Oscillators Coupled Via A Nonuniform Environment

In all kinds of complex systems including physics, chemistry, biology and so on , we can observe various spatiotemporal dynamics. Such behaviors are believed to result from the mutual coupling among each elements. One of typical interaction modes is that each element is coupled through an external environment. This thesis mainly investigated the spatiotemporal dynamics in a large population of chaotic oscillators coupled via a diffusive environment, such as synchronization transition, formation of spiral waves and wave competition between spirals and target waves. Specifically, the thesis is organized as follows:In Chapter 1, we introduced the background of studies on spatiotemporal dynamics behavior and numerical methods. We also stated the problem that we were going to investigate in the thesis.In Chapter 2, we studied the synchronization behavior in a population of Rossler chaotic oscillators coupled via a diffusive environment. In the case of zero dimension, We found that the system would experience a series of period-doubling bifurcation as functions of the coupling strength and density. In one dimensional case where lots of Rossler chaotic oscillator are arranged in a straight line, we found a nontraditional synchronous transition. That is, given the coupling strength, firstly increasing the density will lead to the transition from the stationary state to synchronous oscillation. However, further increasing the density would lead to synchronization from the same initial conditions. We presented synchronous phase diagram as a function of the coupling strength and the density. Finally, we briefly discussed the frequency synchronization for non-identical systems and formation of traveling waves for both heterogeneous and homogeneous cases.In Chapter 3, we mainly studied wave pattern formation, competition and selection in a system of a large group of chaotic oscillators coupled through a diffusive environment in the two-dimensional case. We found that in such a case spiral waves would emerge spontaneously, and their dynamics largely depends on the parameter density. After introducing target waves in the system, we found that the spiral waves and target waves are in competition that the density is larger, the spiral waves occupies the entire system; in contrast on the dominant patterns are target waves. The phenomenon can be well explained by two kinds of frequency dependence of waves on the density. By the way, in a state of multiple of spiral waves, a local heterogeneity would cause the formation of large spiral waves that finally swept the other small spirals, which means a transition from disorder to order state. We also checked some results by using FitzHugh-Nagumo model, and found some inconsistent resultsThe Chapter 4 is our summary and outlook of the paper.