Hopf Bifurcation And Turing Patterns Of Gierer-Meinhardt System

In many fields of chemistry and biology,many natural phenomena can be described by the reaction-diffusion equation.In recent,years,the research of bifurcation problem and other related problems of the reaction-diffusion equa-tion is one of the hot topics among scholars.Bifurcation includes two major categories:dynamic bifurcation and static bifurcation.This paper focuses on the dynaimic bifurcation:Hopf bifurcation and Turing instability.Accord-ing to the definition of the Turing instability,Turing instability could induce various patterns in this system,that is,Turing patterns.For the study of Turing patterns,this paper mainly explores the amplitude equation which is an important tool to study the reaction diffusion systems.In this paper,the problems of Hopf bifurcation,Turing instability and Turing patterns in Gierer-Meinhardt system are studied under Neumann boundary conditions by mathematical methods such as normal form,center manifold theorem,stability theorem and multi-scale analysis.The main results are as follows:1.The condition of Hopf bifurcation was investigated and the direction of the Hopf bifurcation by normal form was determined in Gierer-Meinhardt system without diffusion.Whether the system could generate limit cycles from the Hopf bifurcation was also discussed in this paper.2.The influence of diffusion terms on the stability of equilibrium points and the bifurcated limit cycle from the Hopf bifurcation of Gierer-Meinhardt system were explored in this paper,and the results were verified by numerical simulation.3.Amplitude equation for two-dimensional Gierer-Meinhardt system was derived by using multi-scale analysis.The conditions of the selection of pattern were obtained by analyzing the existence and stability of the solutions of amplitude equation.Finally,the above theoretical results were verified by numerical simulation.