Bifurcation And Pattern Formation Of A Class Of Reaction-Diffusion System

In recent years,the nonlinear problem is one of the heat subject that attracted scholars in the reaction diffusion systems.Bifurcation and Turing instability are the important basis for the study of pattern formation in the dynamics,and the amplitude equation is an important method for the study of the pattern formation of the specific Turing pattern.In this paper,we use dynamical techniques,such as stability theory,nor-mal form,center manifold theory and multi-scale analysis to illustrate the Hopf bifurcation,Turing instability and pattern formation about Gierer-Meinhardt activator-substrate model under Neumann boundary.Details are as follows:1.The stability of the equilibrium points of the system without diffusion are researched,the condition and bifurcation direction of Hopf bifurcation about the equilibrium points are discussed.2.The effects of diffusion on the stability of equilibrium points are consid-ered,the bifurcated limit cycle from Hopf bifurcation are researched,and the condit.ions of Turing instability are investigated.To illustrate the theoretical analysis,we carry out the numerical simulations.3.The amplitude equation is obtained by multi-scale analysis.By an-alyzing the existence and stability of the solution of the amplitude equation,and then the specific pattern formation is obtained.The numerical simulations shows the theoretical analysis.