Study Of Bifurcation Problems Of Two Classes Of Planar Filippov Systems

In this thesis, we mainly discuss bifurcation problems of two kinds of planar Filippov system. By using qualitative theory of ordinary differential equations and constructing proper Poincaré map, we first discuss bifurcation problem of a planar Filippov system constituted by two linear systems with different dynamics, and then we discuss bifurcation problem of a planar Filippov system constituted by a linear system and a cubic Hamiltonian system. To be concrete, we have done the following two parts of research work.In the first part of this thesis, the planar Filippov system we consider has five parameters. Its left subsystem will appear saddle or node and its right subsystem will appear focus when the parameters vary. We obtain that two limit cycles can bifurcate from this system. Moreover, we investigate the sliding bifurcation phenomena. To our knowledge, there are few results on the sliding bifurcation and pseudo homoclinic bifurcation phenomena when the left subsystem and right subsystem have different dynamics.In the second part of this thesis, the planar Filippov system we consider has four parameters. It is constituted by a linear system and a cubic Hamiltonian system. Its left subsystem will appear saddle, node or focus when the parameters vary. We realize that when the left linear subsystem has a saddle or node, the planar Filippov system can have one periodic orbit. There exists no sliding bifurcation phenomenon.However, if the linear system has a focus, two limit cycles can appear in the planar Filippov system. Moreover, there exist sliding bifurcation phenomena. To our knowledge, there are few results on the bifurcation problems of planar Filippov system constituted by a linear system and a cubic Hamiltonian system.