The Research Of Bifurcations Of A Class Of Nonautonomous Reaction-diffusion Equations

Bifurcation problem is one of the most popular research topics in infinite dimensional dynamical systems,which is of great theoretical significance and practical values.This paper is aimed at the research of transcritical bifurcation and pitchfork bifurcation in a class of nonautonomous reaction diffusion equations by using the mathematical theorem and method of semigroup of operators,nonautonomous dynamical system and the bifurcation theorem of infinite dimensional dynamical systems.Innovative results are as follows:1.A class of attractive and repulsive conditions on trivial solution of nonautonomous reaction diffusion equations are given by using the semigroup of operators theorem and nonautonomous dynamical system theorem,which provides methods and approaches of study on transcritical bifurcation and pitchfork bifurcation of this kind of equation.2.The transcritical and pitchfork bifurcations for a class of nonautonomous reaction diffusion equations are given by combining nonautonomous system.According to the theorical results,we know that when the systemic parameters change,the attraction radius of this system becomes zero on appropriate conditions,which means transcritical bifurcation happens in the system.Otherwise,as the systemic parameters change on other conditions,we know that pitchfork bifurcation happens in the system.We extend both the transcritical bifurcation theorem and the pitchfork bifurcation theorem which are applicable to autonomous reaction diffusion equations to these theorems which can be used for nonautonomous reaction diffusion equations.The results we obtained will allow for the clearer understanding of the mechanism of the formation of both the transcritical and pitchfork bifurcations for nonautonomous reaction diffusion equations.