Bifurcation Analysis For Several Classes Of Discontinuous Differential Systems

This thesis deals with the bifurcations of several classes of planar discontinuous differential systems.Firstly,by using the Filippov theory,we discuss the existence and stability of equilibria in the discontinuous differential systems,especially,the singularities on the switching boundary.With the qualitative theory of ordinary differential equation,the properties of the half-return maps of each subsystem are obtained,accordingly,we can establish some results on the sliding limit cycles,homoclinic connections and heteroclinic connections.Additionally,the suitable Poincare map or displacement function can be constructed to explore the existence and stability of the crossing limit cycles,and then plot the complete bifurcation diagram.The main work of this thesis includes the following aspects:1.We describe the global phase portraits for a general planar piecewise linear system separated by a straight line,where two subsystems both having a real focus.In order to achieve the goal,we establish the concrete conditions for the existence of the equilibria(including the equilibria at infinity and pseudo equilibria on the switching boundary),crossing limit cycles,sliding limit cycles and homoclinic connections.Based on these results,we can obtain the critical conditions of the pseudo-homoclinic bifurcation,saddle-node bifurcation of periodic orbit,sliding heteroclinic bifurcation,critical sliding bifurcation and crossing sliding bifurcation,which would facilitate the complete bifurcation diagram.Finally,numerical examples confirmed our theoretical results.2.We dedicate to the sliding bifurcations of planar piecewise smooth system separated by an ellipse.A general planar piecewise smooth system with a sliding heteroclinic loop,a planar piecewise linear system with the center of the ellipse as the unique equilibrium,and a planar piecewise smooth system with nonlinear subsystems have been discussed respectively,and obtained some new phenomena that are not available when the switching boundary is a straight line,such as,a crossing limit cycle containing four intersections with the switching curve,sliding cycles having four sliding segments,and sliding cycles consisting of the entire switching curve and boundary limit cycle.3.The non-smooth bifurcation around a class of critical crossing cycles,which are codimension-2 closed orbits composed of tangency singularities and regular orbits(a loop homoclinic to a fold-fold singularity,or,a heteroclinic loop connected with two regular-fold singularities),for a two-parameter family of planar piecewise smooth system with two zones has been considered.Under the small perturbations of parameters,we obtain the critical conditions of the appearance and disappearance of crossing limit cycles and sliding limit cycles near those loops.Finally,the complete bifurcation diagrams are described,and several examples are given to illustrate our main conclusions.4.We deal with the periodic orbit bifurcations of a T-periodic perturbed piecewise smooth system whose unperturbed part has a generalized heteroclinic loop connecting a hyperbolic critical point and a quadratic tangential singularity.By constructing several displacement functions that depend on perturbation parameter ε and time t,sufficient conditions of the existence of a homoclinic loop and a sliding generalized heteroclinic loop(that is a generalized heteroclinic loop a part of which lies on the switching manifold)are obtained.As the application,we give a concrete example to show that under suitable perturbations of the generalized heteroclinic loop the corresponding phenomena can appear.