Dynamical Behavior Of Several Differential Equations And Its Application To Burgers-Fisher Equation

This thesis focuses on the dynamic behavior of several types of differential equations,including the global structure of the system,limit cycles and the existence of periodic wave solutions.This thesis is divided into the following seven chapters.In Chapter 1,we introduce the background knowledge and main results of this thesis.In Chapter 2,we study a class of four-region piecewise linear systems with three switching lines.The finite singularities,infinite singularities and the number of limit cycles of the system with partial parameters are studied.In Chapter 3,we classify the topological phase diagrams of 2q-3 degree Zqequivariant plane Hamiltonian systems.In addition,we prove that for any 2q-3 degree Zq-equivariant plane Hamiltonian systems(for fixed q∈N,q≥3)at least q limit cycles.In Chapter 4,we consider a class of generalized Burgers-Fisher equations.When m=1,3 2m=k-2(k>1,k is integer),we prove that the generalized Burgers-Fisher equation admits an isolated periodic wave solution based on the monotonicity of the ratio of two Abelian integrals.In Chapter 5,we investigate the global existence and uniqueness of the isolated periodic wave solutions of the Burgers-Fisher equation in another general form.When l=2(n=1,δ<0);l=1(n=2,δ<0);l=1(n=3);l=2(n=2);l=3(n=1),we get the system has an isolated periodic solution.In Chapter 6,we study the persistence of kink and anti-kink wave solutions for the perturbed double sine-Gordon equation by using the geometric singular perturbation theory and the Melnikov method.We give the explicit expression of Melnikov function.In fact,the monotonicity of period function for unperturbed double sine-Gordon equation is investigated.In Chapter 7,we summarize the contents of the thesis and look forward to future research work.