Limit Cycle And Bifurcations In Planar Filippov Systems

The qualitative theory of discontinuous differential systems has become one of the most active research fields in dynamical analysis.This is mainly due to the fact that such systems can be widely used to simulate discontinuous events in the real world,such as the switching of circuit systems,the impact of mechanical devices,the activity of neurons,the vibrations of oscillators with dry friction,the threshold control in pest prevention and cure,etc.Among discontinuous differential systems,the most studied category is discontinuous piecewise smooth differential system,also called Filippov system.It consists of several smooth ordinary differential equations defined on different areas that are separated by some smooth curves or surfaces,named switching boundaries.Similar to smooth systems,the study for limit cycles of Filippov systems is also an important and challenging subject.Due to switching boundaries,the study for limit cycles of Filippov systems mainly focuses on two novel types of limit cycles,i.e.,crossing limit cycle and sliding limit cycle.In Chapter 3 we study the limit cycles of planar discontinuous piecewise linear systems with a line of discontinuity(DPWLL systems).Although it is the simplest type of Filippov systems,the nonlinearity caused by discontinuity still leads to great difficulties in analysis.At present,a great deal of work has been devoted to the study of the maximum number of crossing limit cycles for such systems,but this problem has not yet been solved.Unlike these efforts,in this thesis we study not only the maximum number of crossing limit cycles,but also the maximum number and topology of sliding limit cycles,and coexistence between two types of limit cycles.First of all,we prove that the considered DPWLL system has at most two sliding limit cycles,and that a single(resp.two)sliding limit cycle has 4(resp.3)different topological configurations in the sense of Σ-equivalence.Depending on these different configurations,we further demonstrate that two sliding limit cycles coexist at most with one crossing limit cycle,and one sliding limit cycle can coexist with two crossing limit cycles.In addition,by promoting a uniqueness theorem on the limit cycles of Lienard systems,we also prove that the DPWLL systems without sliding motion allow at most one limit cycle and this number can be reached.Applying this result,we give a positive answer to the conjecture in[Progress and Challenges in Dynamical Systems,Springer,2013,P.232]about the uniqueness of crossing limit cycles for DPWLL systems of focus-focus type without sliding motion.Finally,we make a comprehensive analysis for the limit cycles of the DPWLL systems with the same coefficient matrix,and come to the conclusion that the maximum number of both crossing limit cycles and sliding ones is 2,but the maximum number of coexistence limit cycles is 3.Bifurcation analysis is another important subject in differential systems.Due to switching boundaries,Filippov systems have some typical limit cycle bifurcation phenomena that do not exist in smooth systems,such as pseudo-Hopf bifurcations,sliding bifurcations,etc.In this thesis we focus on a particularly interesting type of sliding bifurcations,i.e.,grazing-sliding bifurcation,generally speaking,which occurs when a hyperbolic limit cycle of one subsystem of the unperturbed system is tangent to the switching boundary at a regular-fold point.In Chapter 4 we contribute to a class of degenerate grazing-sliding bifurcations,assuming that this regular-fold point degenerates to be a fold-fold point.First we prove that the bifurcations are of codimension-two.Then,depending on the type of this fold-fold point,we establish some bifurcation diagrams,as well as the asymptotics of all co-dimension one bifurcation curves.Our work not only reports some of complete unfoldings in the investigation of degenerate grazingsliding bifurcations,but also answers an open problem raised in[J.Differ.Equ.255(2013),4403-36]on the maximum number of limit cycles bifurcating from a generalized homoclinic orbit.