Limit Cycle Bifurcations For Piecewise Smooth Systems

As well known,it is challenging to obtain the maximal number of periodic solutions or an upper bound of it for planar differential systems,which is related to the second part of Hilbert’s 16th problem.In recent years the bifurcation of limit cycles of continuous and discontinuous differential systems has been extensively studied by many authors.The bifurcation theory,which considers the changes of orbital behavior in the phase space as the parameters in a given system vary,has been developed from smooth systems to non-smooth systems.There are two main methods of studying bifurcation of limit cycles or periodic orbits:the Melnikov function method and the averaging method.It is worth noting that the averaging method is equivalent to the Melnikov function method for studying the number of limit cycles of planar analytic(or C∞)near-Hamiltonian systems.In this thesis,we discuss the limit cycle bifurcations using the two methods and develop the related theory about two methods.This thesis consists of the following three chapters:In Chapter 1,we introduce the background and main topics of our research.Also,we describe the structure of this thesis.In Chapter 2,we study the maximal number of limit cycles for a class of piecewise smooth near-Hamiltonian systems under polynomial perturbations.Using the second order averaging method,we obtain the maximal number of limit cycles of two systems respec-tively and present an application.In Chapter 3,we investigate the bifurcation methods of limit cycles for piecewise smooth systems.We first concern the Melnikov function method and averaging method of perturbed piecewise integrable systems in arbitrary dimension,and establish a relationship between them by Poincare map.Then we present the form of the second order Melnikov function for planar perturbed piecewise Hamiltonian systems.Finally we use the results obtained to give an application.