Limit Cycle Bifurcations In Some Smooth And Non-smooth Systems

In this paper, we mainly investigate bifurcation of limit cycles in some smooth andnon-smooth systems.In Chapter one, we introduce the background of our research and main topics thatwe will study in the following chapters.In Chapter two, we study a quadratic system with a global center under polynomialperturbations of degree n (n≥1). By using the first order Melnikov function, we studyPoincaré and Hopf bifurcations. We prove that both Poincaré and Hopf cyclicity are n forn≥2up to the first order in ε.In Chapter three, we consider a class of discontinuous Li′enard systems and study thenumber of limit cycles bifurcated from the origin when parameters vary. We establish amethod of studying cyclicity of the system at the origin. As an application, we discusssome discontinuous Li′enard systems of special form and study the cyclicity near the origin.In Chapter four, we mainly concern limit cycle bifurcations by perturbing a piecewiselinear Hamiltonian system. We first obtain all phase portraits of the unperturbed systemhaving at least one family of periodic orbits. By using the first order Melnikov function ofthe piecewise near-Hamiltonian system, we investigate the maximal number of limit cyclesthat bifurcate from a global center up to first order of ε.