Limit Cycles For Two Classes Of Planar Piecewise Smooth Cubic Differential Systems

In real planar qualitative theory of differential systems,the existence,stability,distribution and the number of the limit cycle are full of important theoretical value and practical significance.Recently,for the researchers,planar piecewise smooth differential system has caused considerable interest,which has been from mechanics and electronic engineering theory and automatic control theory,thus mathematical workers have gained a lot of research work about piecewise smooth differential systems.This paper mainly studies the bifurcation of limit cycles in two classes of planar piecewise smooth differential systems under small perturbations.Comparing with the smooth differential system,there may be a class of non-smooth periodic orbits in phase plane.Using averaging methods of discrete differential system,we discuss that the number of limit cycles can be generated with two different kinds closed orbits after perturbation.After calculation and analysis,it is concluded that the lower bound of the number of the limit cycles from periodic annulus of the origin for the above differential system can be gained.The full text is divided into three chapters,the specific content is organized as follows:The first chapter is the preface.In the first chapter,we introduce research back-grounds of the Hilbert's problem and current situation on smooth differential systems.Furthermore,the research status of piecewise smooth polynomial differential system also has been introduced.We have made a brief description on the result of generalization between the smooth of the differential system about studying the limit cycles and the piecewise smooth differential system.This main work of the article is briefly discussed and the structure of the paper is also given.In the second chapter,we will consider the number of limit cycles in a class of piecewise smooth cubic polynomial differential system with quadratic invariantly curve.More precisely,using the first order averaging method,we can estimate that the least number of limit cycles which bifurcate from any region of periodic annulus of(1.6)?=0.In the third chapter,we will consider the number of limit cycles in a class of piecewise smooth cubic polynomial differential system with quadratic-intersecting invariantly curve.More precisely,using the first order averaging method,we can estimate that the least number of limit cycles which bifurcate from any region of periodic annulus of(1.7)?=0.