The Stability,Turing Instability And Hopf Bifurcation Of Brusselator Model

In recent years,one of the important research topics in dynamical systems has been widely used in mechanics,chemistry,biology,and social sciences and economics etc.The specific evolutionary process of a large number of phenomena has been abstractly referred to as the reaction diffusion equation.In this paper,by using the linearization method and analyzing the distribution of the roots of the corresponding eigenvalue problem on the complex plane in detail,we mainly analyze the stability and Turing instability of the ordinary differential system and the corresponding partial differential system with the homogeneous Neumann boundary condition of Brusselator system:(?)Using MATLAB software package and numerical methods for solving partial differential equations,some numerical simulations to verify theoretical conclusions are given.The first chapter summarizes the research background and current status of the Brusselator reaction-diffusion model,points out the main content and conclusions of this paper,and briefly introduces the concepts needed in the article.The second chapter considers the local asymptotic stability and Hopf bifurcation of the positive equilibrium point of the ordinary differential system corresponding to the Brusselator reaction-diffusion model.The MATLAB software package is used to give numerical verification of the theoretical results obtained.The third chapter discusses the Brusselator reaction-diffusion model under the Neumann boundary.When there is a diffusion term,we study the linearization of the system at the spatially homogeneous positive equilibrium point and analyze the corresponding eigenvalue equation to prove the system's normal number equilibrium solution Asymptotic stability and Turing instability,and corresponding numerical simulations are given for the theoretical results obtained.Chapter 4 considers the Hopf branch of the Brusselator reaction-diffusion model and the stability of the Hopf branch's direction and the periodic period solution,and uses the canonical theory and central flow theorem to analyze the branch direction and the periodic period solution of the homogeneous Hopf branch in the diffusion system space.In order to verify the correctness of the theoretical conclusions,numerical verifications are given for some specific examples.