The Dynamic Behavior Of Patterns In Reaction-diffusion System

In nature, patterns are the spatiotemporal periodic structure widespread. Patterns are self-organized in conditions far away from thermodynamic equilibrium. Patterns are widespread in nature, therefore the research on pattern dynamics has a high reference value. In the field of Nonlinear Science, Scientists have been trying to research the reasons and rules of the pattern's form from the perspective of dynamics, Nonlinear dynamics Pattern study becomes an important branch of nonlinear science. In all kinds of pattern, spiral wave is widespread and has universal dynamic behavior. Research and master these rules has great potential application value. Reaction diffusion system is one of systems that able to produce the spiral wave, The complex Ginzburg-Landau equation(CGLE) is a model frequently used for the study of spatiotemporal system, and is one of the common models of reaction-diffusion system. In this work, we consider the complex Ginzburg-Landau equation(CGLE) as the model of reaction-diffusion system to study complex dynamic behaviors of spiral waves and the evolution of target waves based on spiral waves.The first chapter serves as a brief introduction of pattern and pattern dynamics, the formation and characteristics of spiral wave. Then, the reaction-diffusion system and CGLE is introduced, and the stability of traveling-wave solutions about CGLE has been analyzed.The second chapter investigates the pattern dynamics in unidirectionally coupled spatiotemporal systems based on a model of complex Ginzburg-Landau equation. Driven by a stable amplitude spiral waves system, the response system with different coupling strength is discussed. According to the analysis of the synchronization function and frequencies of driving and response systems, similarity of spiral structures between phase and amplitude is observed under weak coupling conditions. The same phenomenon is also found with the driving signals of target wave and plane wave, respectively. This proves the process of synchronization in unidirectional coupled system has three steps---from out of synchronization to similarity of spiral structures between phase and amplitude, and finally to complete synchronization. This phenomenon can explain the occurrence of amplitude spiral.The third chapter works on two kinds of target wave generation mechanism. Based on stable spiral wave, we set system under a local negative feedback and a local oscillation to make the system produces a target wave source, these two kinds of target wave both can make the system original spiral waves system being effective controlled. And the control effect of the oscillation control method is relative to oscillation parameter. We find a optimal oscillation parameter after a lot of experiments. With the the competition of waves, we compared the two kinds of target wave's transmission stability. Results showed that the target wave with a local negative feedback has higher stability, the ability of controlling spiral waves was better. This conclusion provides the more effective theory for controlling the spiral wave.The fourth chapter discusses oscillatory frequencies in spatiotemporal system with local inhomogeneity. In this work, we investigate the spiral pattern evolution in CGLE, and observe that the spirals can be changed to be stable target waves with local inhomogeneous parameter shifts in the system space. A successful transfer can be achieved when the oscillatory frequencies of non-controlled space and the local inhomogeneous area, which have equal values, are both smaller than the eigen-frequency of the system, and an intriguing V-shaped line is found in parameter-frequency diagram. Some interesting features of the V-shaped line are shown in the further elaborated studies in this work.The fifth chapter is the summary of the whole thesis.