A Periodic Orbital Branch Of A Three-dimensional Piecewise Smooth System

In this paper,we consider a three-dimensional piecewise smooth system having an invariant surface and a closed orbit of multiplicity k on the invariant surface.By using bifurcation techniques and analyzing the solutions of bifurcation equations,we study the special bifurcation phenomena near the multiple k closed orbit and obtain some conditions for the existence of periodic orbits bifurcated from it.As an application,we study the number of limit cycles for a concrete piecewise system.In addition,we also study the multiplicity of closed orbits appearing in Poincare bifurcations of planar piecewise Hamil-tonian system.This eaasy is composed of the following six chapters:In Chapter 1,we briefly introduce the research background and give the definition of multiple k closed orbit of planar piecewise smooth system.In Chapter 2,we give some basic definitions related to this paper and also introduce some known results as lemmas.In Chapter 3,we deal with the three-dimensional piecewise smooth system by apply-ing curve coordinate transformation.Furthermore,we obtain the bifurcation equations of the system.In Chapter 4,we study the limit cycles of the three-dimensional piecewise smooth system having an invariant surface by analyzing the solutions of bifurcation equations.Furthermore,we obtain the sufficient conditions of the existence of limit cycles.In Chapter 5,we study the multiplicity of closed orbits appearing in Poincare bifur-cations of planar piecewise Hamiltonian system.We give upper bounds of the multiplicity of the closed orbits for the case that the Hamiltonian function is a piecewise polynomials of high order.In Chapter 6,as applications,we study the number of limit cycles for a concrete three-dimensional piecewise smooth system having an invariant surface and a closed orbit of multiplicity 2 on the invariant surface.We also study the max multiplicity of a closed orbit of a concrete planar piecewise system.