Stability And Bifurcation Of Several Classes Of Systems With Reaction-diffusion

Reaction-diffusion systems, as one of the basic equations which can describe the motion of nature,usually solve the nonlinear problems and have rich varieties of dynamical behaviors, in which thebifurcation is one of the basic features. As advanced topics in Nonlinear Science, bifurcation withprofound application background still remains open and challenging. In this thesis, based on theanalysis and summary of research status of reaction-diffusion systems, employing partial functionaldifferential equation theory, stability, bifurcation and Turing’s theory, the author investigates stabilityand bifurcation of several classes of systems with reaction-diffusion. The organization of this papertakes the following form:In the first Chapter of this dissertation, the author elaborate the current status and researchprogress about reaction-diffusion systems, particularly in chemical and neural network systems andexpounds the main contents of this paper.The second Chapter studies2-D Lengyel-Epstein reaction-diffusion system with homogeneousNeumann boundary condition, present some conditions ensuring the equilibrium of system to bestable and derive conditions on the parameters so that spatial homogenous Hopf bifurcation andTuring instability occur. These conditions obtained have important leading significance in applicationof L-E system.The third Chapter deals with the synchronized stability and Hopf bifurcation of a class of ringneural networks with leakage time-lags and reaction-diffusion, determiningτandβas thebifurcation parameter respectively, give some criteria for synchronized stability and Hopf bifurcation,also give the conditions of stability and Hopf bifurcation for the corresponding system withoutreaction-diffusion.The fourth Chapter discusses the stability and Hopf bifurcation of a class of a neuron model withreaction-diffusion and delay-dependent parameters, determiningτas the bifurcation parameter, givesome criteria for stability and Hopf bifurcation. It shows that controllers which are designed in thischapter can control the occurrence of bifurcation effectively and can dominate the amplitude of thebifurcation limit cycle.The fifth Chapter summarizes the research work of this dissertation. Furthermore, the futureresearch direction is made.