Bifurcation Problem In Turing Pattern Dynamics

Pattern is a kind of regular and nonuniform macrostructure in space or time. There are various kinds of pattern structures, which form a colorful world. There-fore, learning the cause and mechanism about the formation of pattern, is of great significance to uncover the mystery of nature for the formation. In 1952, the fa-mous British mathematician Alan Mathion Turing published his famous paper The chemical basis of morphogenesis’",a reaction-diffusion model was used suc-cessfully to illustrate some of the patterns that appear on the surface,such as zebra pattern is how formed. With the development of the times, this theory has given rise to chemistry, physics, mathematics, biology and other disciplines researchers interests and attentions.The mathematical description of the general mechanism of Turing patterns in dynamic systems is that the stable constant equilibrium solution of ordinary differential system occurs stability reversed after the addition of diffusion, so that it will produce Turing pattern in the vicinity.The introduction in the first, section describes the specific sources of back-ground of pattern dynamics theory, and gives Lyapunov stability theory in the second quarter. Then,in the third quarter and the fourth quarter,the condi-tions arising Turing instability of free diffusion system in one-dimensional and n-dimensional space were given. And in the last part of the introduction, we gave some preparing knowledge for our theoretical research about bifurcation. Most of the current studies only done some research on when the bifurcation of the system occurring Turing unstable appears. But neither has the bifurcation curve been deeply studied,nor has the stability of bifurcation solution been explored. So,this paper will mainly study on these two issues. In Chapter 2,according to Crandall-Rabinowitz theory,we study the the bifurcate condition when the one-dimensional free diffusion system occurred Turing unstable.And we also explore the local bi-furcation solutions at the bifurcation point,then we give the corresponding local bifurcation diagrams.And in the last quarter of Chpter2,we describe in details of the stability of local bifurcation solution.In chapter 3, we discuss whether the bi-furcation in the n-dimensional free diffusion system occurs at first.Then,we mainly study in the case that the free diffusion appear in a two-dimensional rectangular area as an example.we analyse the bifurcate conditions,and study deeply on the local bifurcation curves at the fist eigenvalues. Finally,Chapter 4 summarizes the information given above.