Bifurcation Analysis In Some Piecewise Smooth Systems

In this thesis, we mainly investigate the bifurcations on several classes of piecewise smooth systems after perturbations. Compared with smooth systems, there can exist a new type of singular point, usually called generalized singular point. According to the type or location of the singular point, in phase plane there may exist a special periodic orbit,non-smooth even discontinuous. By applying the first order Melnikov function and taking proper variable transformations, we discuss the lower bound of maximal number of limit cycles.This thesis is composed of the following five chapters.Chapter 1 is devoted to the introduction concerning the background, preliminary and the method used in this paper.In Chapter 2, we consider a class of linear piecewise Hamiltonian systems with one or two hyperbolic saddles and the origin being a generalized center in phase plane. Taking some variable transformations, the first-order Melnikov function can be expanded in the bounded domain and then we discuss Hopf, Poincar′e, homoclinic and heteroclinic bifurcations in order to complement the unsolved cases of the 13 di?erent conditions presented in[53].In Chapter 3, we investigate a class of piecewise Li′enard systems with some small perturbation, where the origin is a generalized or hyperbolic saddle. After expanding the first-order Melnikov function, we study the number of limit cycles. We su?ciently utilize the relationship between simple zeros and limit cycles and give lower bounds of the maximal number.In Chapter 4, we consider a special piecewise system with a double homoclinic loop and limit the critical point. It is proved that there are three families of closed orbits before perturbing. We obtain three Melnikov functions and some relationship between the coe?cients of the expansions with respect to Melnikov functions. Then we obtain the new results concerning the number of limit cycles.In Chapter 5, we analyze a class of piecewise polynomial systems, which contains a higher order singular point and an elementary singular point. We prove there exist 17 cases with di?erent conditions and phase portraits on the unperturbed system. In particular, we describe a special case with a heteroclinic loop and illustrate the result about the number of limit cycles.