Hopf Bifurcation And Neimark-Sacker Bifurcation For A Class Of Rayleigh Equation

For a nonlinear dynamical system, when its parameters through some critical value, the structure of the solution set of the system may change heavily. This is the problem the bifurcation theory of dynamical systems to be studied. In this dissertation, we consider the dynamical behavior of a class of Rayleigh equation in both cases of no excitation and small excitation. Firstly, we give the conditions of occurring the Hopf bifurcation and Neimark-Sacker bifurcation for the system. Then we use numerical simulation to verify the theoretic results.In Chapter 1 we survey the present situation and some achievements of the bifurcation theory. Moreover, we introduce the main results of this dissertation.In Chapter 2 we recall some basic results of the bifurcation theory which will be used in what follows, which include Hopf bifurcation and Neimark-Sacker bifurcation.In Chapter 3 we study Hopf bifurcation of periodic solutions of a class of Rayleigh equation. First we transform Rayleigh equation into a two-dimensional differential equation, and by the coordinate transformation, write differential equation in the polar coordinate form. Then we use the normal form theory to determine the conditions of Hopf bifurcation of the system. Finally we verify the theoretical results by numerical simulations.In Chapter 4 we study the dynamical behavior of a class of Rayleigh equation under small periodic excitation. Firstly, we change the original equation into the polar coordinate form and use series method to obtain the approximate solution. Moreover, we construct a Poincare map for the system, Then we use the normal form theory to determine the conditions of Neimark-Sacker bifurcation of the systems. Finally we verify the theoretical results by numerical simulations.