Bifurcation Analysis Of Planar Piecewise Smooth Differential System With Tangent Points

In this thesis we provide a bifurcation analysis for a planar piecewise smooth differential system with tangent points.This system has 3 parameters and it consists of a linear system and a quadratic Hamiltonian system.When parameters vary,the equilibrium point of the linear system can be a saddle,a focus or a node.The quadratic system has some tangent points on the discontinuity line.If the linear system has a focus,we prove that the piecewise smooth differential system will have a periodic orbit and a sliding cycle.Moreover,this piecewise smooth system will undergo pseudo-homoclinic bifurcation and critical crossing bifurcation CC.If the linear system has a node,we also find that this piecewise smooth system will have a sliding cycle and undergo pseudo-homoclinic bifurcation and critical crossing bifurcation CC.As far as we know,the results that we see a sliding cycle,pseudo-homoclinic bifurcation and critical crossing bifurcation in a planar piecewise smooth system with a node are new.Finally,some examples are given to verify our results.The results in[E.Freire,E.Ponce,E.and F.Torres.On the critical crossing cycle bifurcationin planar Filippov systems.J.Differential Equations,259?2015?,7086⁷107]show that critical crossing bifurcationCC₂ will not occur in the codimention 1 bifurcation problem of planar piecewise smooth systems.Our results show that critical crossing bifurcationCC₂ will not occur in planar piecewise smooth systems with multiple parameters.