The Turing-Hopf Bifurcation Of A Gierer-Meinhardt Model

The bifurcation theory of reaction diffusion system is one of the hot topics in the field of applied mathematics.The study of bifurcation theory is of great importance in the interpretation of biological and chemical phenomena.Bifurcation includes the bifurcation of codimension 1 and the bifurcation of the high codimension.In this paper,based on the Neumann boundary conditions,we mainly consider the Hopf bifurcation of codimension 1,Bautin bifurcation of codimension 2 and the Turing-Hopf bifurcation of codimension 2 in a GiererMeinhardt system.The main content is arranged as follows:In chapter one,we summary the research background and research actuality of Gierer-Meinhardt model.Moreover,the main contents and arrangement of the research are given.In chapter two,we first investigate the existence about Hopf bifurcation in the Gierer-Meinhardt system without diffusion terms.Then,with the aid of Maple,center manifold theorem and normal form,the first,second Lyapunov coefficient and the universal unfolding of Bautin bifurcation are calculated,the existence of Bautin bifurcation is determinated.Next,we study the existence of Turing instability and the Turing-Hopf bifurcation in the Gierer-Meinhardt system with diffusion terms.Finally,some bifurcation behaviors about Hopf bifurcation and Bautin bifurcation of codimension 2 are numerically simulated.In chapter three,the method of the multiple time scale analysis is adopted to derive the amplitude equations of the Gierer-Meinhardt system in the Turing-Hopf bifurcation point.The solutions of the amplitude equations is further discussed,we find that the Gierer-Meinhardt model may show the spatial,temporal or the spatiotemporal patterns,such as the nonconstant steady state,spatially homogeneous periodic solutions and the spatially inhomogeneous periodic solutions.Finally,some numerical simulations are presented to demonstrate the applicability of the theoretical results we get.In chapter four,the research work of this paper is discussed and summarized,and the prospect of future work is given.