Zero-Hopf Bifurcation Analysis In Two Types Of Nonlinear Models

The aim of this paper is to research the dynamical behaviors of two types of nonlinear models,which are CSTR(continuous stirred tank reactor)model and FHN(FitzHugh-Nagumo)model.And we consider about the Zero-Hopf bifurcations of these systems.Both of the center manifold method and the normal form method are used to reduce the systems.For the research of the CSTR model,we first analyze the existence conditions of Zero-Hopf bifurcation according to Zero-Hopf bifurcation theory,and then calculate the normal form of Zero-Hopf bifurcation through center manifold theory and normal form method,and the changes of local topology caused by bifurcation are discussed.Finally,by selecting appropriate parameters for numerical simulation,we obtain the figures of stable and unstable periodic solutions.In research with reaction-diffusion system dynamics properties of FHN model,we first calculate the Hopf bifurcation line,then calculate the Turing instability borderline,and obtain the conditions of parameter value for the system to undergo the Turing-Hopf bifurcation(namely the special Zero-Hopf bifurcation).Secondly,by using the method of Song yongli et al.to calculate partial differential equations,we introduce two perturbation parameters and obtain the normal form by reducing the system.Finally,the correctness of our theoretical analysis results is verified by numerical simulation.