Turing Instability And Bifurcation Analysis Of A Kind Of Reaction-diffusion Chemical Problem

Bifurcation phenomenon means when the parameters change,the structural properties in the system change.Bifurcation problems mainly include three parts: local bifurcation problems,semi-local bifurcation problems and global bifurcation problems.The study of bifurcation phenomenon has very important practical significance.By using center manifold theory,normal form methods and local Hopf bifurcation theorem for the semilinear partial differential equations,a kind of Boissonade model describing the formation of Turing patterns in chemical reaction is investigated.To this model,we mainly study the Hopf bifurcation problems and Turing bifurcation problems.The main contents are as follows:Firstly,we discuss the existence of the positive equilibrium solution in this system,and study the stability of the positive equilibrium solutions when the parameters in different ranges.By using center manifold theory,we discuss the stability of the ordinary equilibrium solution under a certain conditions and obtain the existence conditions of Hopf bifurcation.We discuss the property of the ordinary equilibrium solution and the positive equilibrium solution under a certain conditions respectively,namely confirm the equilibrium solution is saddle point,nodal point,focal point,centre,or degenerated nodal point.We draw up the bifurcation diagram in the parameter plane under this two circumstance.Secondly,by using Hopf bifurcation theorem for the semilinear partial differential equations,we confirm the bifurcation direction and the stability of the bifurcating periodic solutions near the positive equilibrium solution under a certain conditions.Finally,we obtain the Turing instability conditions of the positive equilibrium solution.