Dynamics And Regularization Of A Class Of Filippov Systems

Filippov systems are a class of discontinuous systems.In Filippov systems,the whole phase space can be divided into several different regions,and the dynamic behavior in each region is consistent with the smooth system.In the paper,we study the dynamics of a class of Filippov systems and Filippov systems that have two ordinary equilibrium points.A class of Filippov systems consists of three parts.One part is that Filippov systems have a center,the other part is that Filippov systems have only a center,and the third part is that Filippov systems have a saddle point only.The Filippov systems with two ordinary equilibrium points are a class of Filippov systems which have a center and a saddle point.This class of Filippov systems whose whole phase space can be divided into two parts by a discontinuous boundary.One vector field is transversal to the discontinuous boundary,and there are three types of boundary equilibrium bifurcations in Filippov systems.Such as Persistence bifurcation,Non-smooth fold bifurcation and degenerate bifurcation.We also show different bifurcation diagrams by Mathematica.Besides,we study the Filippov systems that have a saddle point and center and they have four discontinuous boundaries.We can discuss the dynamics of the regularized systems of Filippov systems by regularization method.This article is divided into six parts:The first chapter is the introduction,which describes the research background,research status and main results of this paper.The second chapter is the preliminary knowledge,gives some relevant theoretical basis of this paper.The third chapter can be divided into three parts.The first part is the dynamic analysis of Filippov systems that have a center.In this part,we obtain four types of bifurcations,such as F-C-LC bifurcation,Persistence bifurcation,Non-smooth fold bifurcation and degenerate bifurcation.The second part is the dynamic analysis of Filippov systems that have only a center.The third part is the dynamic analysis of Filippov systems that have a saddle point only,where we can get three types of boundary equilibrium bifurcations.In Chapter 4,we study the dynamics of a Filippov system with two ordinary equilibrium points,and we get three types of bifurcations in Filippov systems.In Chapter 5,we study the bifurcation of Filippov systems that have two discontinuous boundaries and we can use the regularization method to analyze the dynamics of the whole system.In Chapter 6,we summarize the results of this paper and prospects for the future.