Bifurcation Of Limit Cycles Of Some Planar Poynomial Systems And Piecewise Smooth Systems

In this dissertation, we investigate the bifurcation of limit cycles of some piecewise planar Hamiltonian systems and some polynomial systems. By using a successor func-tion we study the Hopf bifurcation of piecewise planar Hamiltonian systems. By using the Melnikov function we study the Hopf bifurcation, homoclinic bifurcation, double homoclinic bifurcation and the cuspidal bifurcation of some polynomial systems under some perturbations. Further, we give the concrete conditions of obtaining limit cycles. We also give some new results of the lower bounds of the maximal number of limit cycles. This paper is divided into seven chapters.As an introduction, in the first chapter we introduce the background of our research and main topics that we will study in the following chapters. We also give a description of our methods and results obtained in this thesis.In the second chapter we study the number of limit cycles appearing in Hopf bifurcation of piecewise planar Hamiltonian systems. For the case that the Hamiltonian function is a piecewise polynomials of a general form we obtain lower and upper bounds of the number of limit cycles near the origin respectively. For some systems of special form we obtain the Hopf cyclicity.In the third chapter we consider bifurcation of limit cycles in near-Hamiltonian systems. A new method is developed to study the analytical property of the Melnikov function near the origin for such systems. Based on the new method, a computationally efficient algorithm is established to systematically compute the coefficients of Melnikov function. Moreover, we consider the case that the Hamiltonian function of the system depends on parameters, in addition to the coefficients involved in perturbations, which generates more limit cycles in the neighborhood of the origin. We apply the results to a quadratic Hamiltonian system and a quadratic integrable non-Hamiltonian system with cubic perturbations and give the maximal number of limit cycles near the origin for these systems.In the fourth chapter we first give some general theorems on the limit cycle bi-furcation for near-Hamiltonian systems near a double homoclinic loop or a center as a preliminary. Then we use these theorems to study some polynomial Lienard systems with perturbations and give new lower bounds for the maximal number of limit cycles of these systems.In the fifth chapter, we study the number of limit cycles of some polynomial Lienard systems with a nilpotent cusp and obtain some new results on the lower bound of the maximal number of limit cycles for this kind of systems.In the sixth chapter, we study the number of limit cycles of some polynomial Lienard systems with a cuspidal loop and a homoclinic loop, and obtain some new results on the lower bound of the maximal number of limit cycles for these systems.In the seventh chapter we study a cubic system and obtain a concrete condition under which the cubic system has 13 limit cycles.