Study Of Bifurcations In Regularized System Of Planar Piecewise Smooth Systems

In this thesis,we mainly study the regularization of piecewise smooth systems with some special topological structure trajectories,such as sliding heteroclinic loop,grazing heteroclinic loop and generalized homoclinic loop.By constructing Poincarémaps near these trajectories and analyzing the fixed points of Poincaré map,we obtain the existence and the stability of limit cycles in the regularized system.In Chapter 2,we study bifurcations in the regularized system of a piecewise smooth system with a sliding or grazing heteroclinic loop.When a piecewise smooth system with a sliding heteroclinic loop undergoes sliding heteroclinic bifurcation,its regularized system can bifurcate at most 3 limit cycles,one of which is a stable limit cycle.When a piecewise smooth system with a grazing heteroclinic loop undergoes grazing heteroclinic bifurcation,its regularized system can bifurcate at most 2 limit cycles,one of which is stable and the other is unstable.The research content in this chapter is the generalization of sliding homoclinic bifurcation to sliding heteroclinic bifurcation in the piecewise smooth system and the supplement to the content of grazing heteroclinic bifurcation in the piecewise smooth system.In Chapter 3,we study limit cycle bifurcation of the regularized system of piecewise smooth systems with a generalized homoclinic loop.For a piecewise smooth system with a visible fold,by constructing Poincaré-Bendixson annulus near the generalized homoclinic loop,we find that its regularized system has a stable limit cycle.For a piecewise smooth system with a two-fold singularity,by constructing Poincaré map near the generalized homoclinic loop,using the existence and the number of fixed points of Poincaré map,we obtain the number of limit cycles in the regularized system.Furthermore,when the two-fold singularity is visible,the regularized system has at most 1 limit cycle,and we also obtain its stability.When the two-fold singularity is visible-invisible,the regularized system has at most 2 limit cycles.