Limit Cycles And Canards In Planar Piecewise Smooth Slow-fast Systems

The research on canard theory of nonsmooth systems is a natural extension of smooth systems,which has been enriched and expanded in recent years.However,there are field blanks about existence of canard solution,asymptotic expression of control parameter and canard cycles in piecewise smooth systems.In this thesis,we first discuss two types of planar piecewise smooth systems,where the right system is a van der Pol equation and the left is a class of linear or quadratic system.Canard phenomenon of these systems are considered.Finally,the multiscale dynamics of a piecewise smooth predator-prey model is studied.This thesis consists of six chapters:In Chapter 1,we introduce the background of singular perturbation systems and research status of canard phenomenon.As well as the research methods,main results and innovation points of this thesis.In Chapter 2,we briefly introduces the basic knowledge of fast slow systems,geometric singular perturbation theory and related concepts of canard problem.In Chapter 3,we investigate the extending slow manifolds near corners of two types of piecewise smooth systems.Moreover,we analyze the dynamics of trajectory around the corner and give the asymptotic expansion of slow manifold by means of blow-up techniques.In Chapter 4,we study the canard solution of two types of piecewise smooth systems.The necessary and sufficient conditions for the existence of maximal canard are obtained by the blow-up method,and the asymptotic expression of the control parameter is given.In Chapter 5,we begin with a qualitative analysis of the van der Pol system,and then use Poincaré-Bendixon theorem and shadow lemma to analyze the existence,uniqueness,stability and position relationship of limit cycles in two types of piecewise smooth systems.In Chapter 6,the piecewise smooth predator-prey model with Allee effect and Holling I type functional response function is studied.Applying geometric singular perturbation theory,some multi-scale dynamical phenomenons are obtained in the case of different positive equilibrium point counts,such as homoclinic orbits and heteroclinic orbits,canard explosion and relaxation oscillation.