Study Of Bifurcation Problems Of A Class Of Planar Piecewise Smooth Systems

In this dissertation, we mainly consider bifurcations of a class of planar piecewise smooth differential systems. The left subsystem is a general linear system and the right subsystem is a quadratic Hamiltonian system. The general linear system has four parameters and a unique singularity. When the parameters vary in different regions, the linear system can have a saddle, a node or a focus. For each case, we provide an extensively qualitative analysis of dynamical behavior for this piecewise smooth system.In the first place, we consider the case when the left subsystem has a saddle. The piecewise smooth systems may have a homoclinic loop and a heteroclinic loop. We get the corresponding conditions that parameters should satisfy. The piecewise smooth system will have at most one limit cycle.Secondly, when the left linear subsystem has a virtual node, we prove that heteroclinic bifurcation phenomenon will occur. Moreover, there will be at most one limit cycle.Finally, if the left linear system has a focus, we will discuss the bifurcation problems of the piecewise smooth systems from two aspects, i.e. real focus and virtual focus. If the focus is a real focus, when the parameters vary we prove that many bifurcation phenomena will occur such as heteroclinic bifurcation, crossing critical cycle bifurcation, sliding homoclinic bifurcation, sliding heteroclinic bifurcation and so on. We can obtain at most two limit cycles. If the focus is a virtual focus, we prove that only heteroclinic bifurcation will occur. In this case, there exists at most one limit cycle.Our results generalize all the results when the left system has a real focus in [Pi D., Zhang X., The sliding bifurcations in planar piecewise smooth differential systems. J. Dyn. Diff. Equat.,(25)2013, 1001-1026]. Moreover, we discuss the cases when the left system has a saddle, a virtual node and a virtual focus. These results are new.