Normal Form And Limit Cycle Bifurcations In Piecewise Smooth Differential Systems

In the past several decades,discontinuous differential systems are constantly ap-peared in various fields of science,such as electronic engineering,electronic circuits with switches,mechanical engineering with impact,mechanical systems with dry fric-tion,control system and so on.Because of the importance of the discontinuous systems in application,the researches on the theory of the dynamics of these systems has at-tracted lots of attentions,and this theory has gone through a great development in the past decades.In this thesis,we study the qualitative theory of piecewise smooth differential equations.The main researches are in three aspects:normal form of planar piecewise smooth differential systems under topological equivalence;averaging theory of arbitrary order for one dimensional piecewise analytic and periodic differential e-quations,and its application to planar piecewise smooth differential system;and the limit cycle bifurcation near generalized homoclinic loop in planar piecewise smooth differential systems.The study of this thesis is divided into three parts:In the first part,we consider the existence of normal form of piecewise smooth differential systems with ?-center under topological equivalence,and limit cycle bifur-cation under perturbation of them.Normal form is one of the most important tools for studying dynamics of dynamical systems.The normal form theory of smooth dy-namical systems and its application has a long history.This thesis focuses on normal form of piecewise smooth differential systems with ?-center.Buzzi,Carvalho and Teixeira[J.Math.Pures.Appl.m2014]researched normal form of nondegenerate ?-center under topological equivalence.The equivalence,called?-equivalence,is defined through homeomorphism,and the transformation can estab-lish a relation only between the orbits of the two vector fields but not the two vector fields themselves.So when we study the dynamics of general nondegenerate ?-center systems under perturbation,we cannot use those of the normal form system because of the absence of regularities of the transformation.Therefore,finding a smooth homeo-morphism is a key point,and is also a difficulty part.This thesis show that a piecewise smooth differential system with ?-center is topologically equivalent to the normal for-m with nondegenerate ?-center by new idea and method,and the homeomorphism between them is piecewise smooth.This result develops and improves the results of[Buzzi,Carvalho and Teixeira,J.Math.Pure Appl.,2014]in four aspects.1?We present the normal form of a piecewise smooth differential system with either nonde-generate or degenerate ?-center uniformly by a new method,but the result of Buzzi et al.just deals with nondegenerate ?-center.2?We find a transformation which is piecewise smooth.3?Our results can deal with limit cycle bifurcation of the original piecewise smooth differential system,however,the result of Buzzi et al.cannot.4?We consider conjugation between the flows of some piecewise smooth differential systems with a ?-center.In the second part,we investigate averaging theory of arbitrary order for one di-mensional piecewise analytic and periodic differential equations,and its application to planar piecewise smooth polynomial differential system.Averaging theory is one of important tools for studying periodic solutions of differential equations.And so far there are many results for averaging theory of smooth differential equations and its applications,whereas the averaging theory of piecewise smooth differential equations has a starting point in recent years.Llibre et al.[Bull.Sci.Math.139(2015),J.Diff.Eqns.258(2015)]exhibited the averaging theory of first and second order for studying periodic solutions of piecewise smooth differential systems.We note that the averaging theories given by them can deal just with the piecewise smooth differential systems whose unperturbed systems are smooth.Llibre and Novaes[Llibre and Novaes,http-s://arxiv.org/abs/1504.03008v1]achieved the first order averaging theory for studying periodic orbits of the piecewise smooth differential systems,whose unperturbed sys-tems have periodic orbits filled with a lower dimensional submanifold.But the authors provided also an example of their theory to piecewise linear differential systems with the smooth unperturbed systems.As we knew,all examples in above papers,as appli-cations of the averaging theories for piecewise smooth differential systems,have smooth unperturbed differential systems.This thesis considers averaging theory of arbitrary order for one dimensional dis-continuous and periodic differential equations.And we provide a planar piecewise smooth differential system as application of the averaging theory,whose unperturbed system is piecewise smooth.The difficulty in studying averaging theory is in which the variational equations of unperturbed systems along the given periodic solutions are piecewise smooth,and the trouble with the application is to find a suitable change of coordinates to transform planar piecewise smooth differential system to one dimen-sional piecewise smooth and periodic differential equation.This result improves the existing ones of piecewise smooth differential systems in two aspects.First,we pro-vide averaging methods of arbitrary order for one dimensional piecewise analytic and periodic differential equations,however,the existing results contain only the averag-ing methods of first and second order.Second,we improve the work of Prof.Jaume Llibre et al.[Llibre and Novaes,https://arxiv.org/abs/1504.03008v1],such that our results can deal with piecewise smooth and periodic differential equations whose un-perturbed equations are piecewise smooth,whereas those of Llibre et al.disposed just the equations with smooth unperturbed equations.In the third part,we study limit cycle bifurcation near generalized homoclinic loop in some planar piecewise smooth differential systems by the first order Melnikov function.Kunze[Sring-Verlag,(2000)]mentioned that the idea can carry over to non-smooth differential systems.Liu and Han[Internat.J.Bifur.Chaos 20(2010)]showed a general formula of the first order Melnikov function in piecewise smooth near-Hamiltonian systems to study the number of limit cycles which are bifurcated from a family of periodic orbits.Here the periodic orbits are formed by regular orbits of the two subsystems which coincide on the discontinuous line.Liang and Han[Chaos Solitons Fract.45(2012)]studied the number of limit cycles bifurcated from a family of periodic orbits near generalized homoclinic loop via the existing first order Melnikov function,and from three families of periodic orbits near generalized double homoclinic loop.We note that the generalized homoclinic loop is a regular periodic orbit of one of the two subsystems which is tangent to the discontinuous line in unperturbed piecewise smooth Hamiltonian systems,and that the generalized double homoclinic loop is a combination of two regular periodic orbits of the two subsystems both of which are tangent to the discontinuous line at the same point.Wei,Liang and Lu[Appl.Math.Comput.243(2014)]researched the number of limit cycles bifurcated from two families of periodic orbits near a generalized homoclinic loop,which is a homoclinic loop of one of the two unperturbed subsystems with a hyperbolic saddle on a switch line.Liu,Han and Romanovski[Internat.J.Bifur.Chaos 25(2015)]discussed the number of limit cycles near generalized homoclinic or double homoclinic loops of piecewise smooth Lienard systems,where the generalized homoclinic loop is a homoclinic loop of one of the two unperturbed subsystems with a hyperbolic saddle on the switch line at which another subsystem is tangent to the switch line or has a center.The double homoclinic loop is formed by a homoclinic loop of a subsystem and a periodic orbit of another subsystem.Xiong[Internat.J.Bifur.Chaos 26(2016)]gave explicit formulas of the first coefficients of the expansion of the first order Melnikov function outside and near generalized double homoclinic loop in piecewise smooth near-Hamiltonian systems,and sufficient condition for 14 limit cycles appearing in the perturbed systems,where the generalized double homoclinic loop is formed by homoclinic loops of the two unperturbed subsystems with a cusp of order 1 at the origin which is located on the discontinuous lineThis thesis focuses also on bifurcation of limit cycles near generalized homoclin-ic loop in piecewise smooth near-Hamiltonian systems,however,here the generalized homoclinic loop can be viewed as a homoclinic loop of one of subsystems with a nilpo-tent saddle or a cusp of arbitrary order.The difficulty is the concrete formulas of the coefficients of the expansion of the first order Melnikov function in the Hamiltonian value h,especially,the formulas of coefficient in h.This result improves and devel-ops the existing results on limit cycle bifurcation of smooth differential systems and piecewise smooth differential systems in two aspects.First,we present the asymptotic expansion of the first order Melnikov function in the Hamiltonian value h,outside and near generalized homoclinic loop with nilpotent saddle or cusp of arbitrary order,and the property of the first coefficients of the expansion.Our results can be regarded as supplement of the existing results on Melnikov function in piecewise smooth differen-tial systems.Second,we improve and supplement the results of Prof.Maoan Han,Ali Atabaigi,Yanqin Xiong et al.[Zang,Han and Xiao J.Diff.Eqns.245(2008),Han,Zang and Yang,J.Diff.Eqns.246(2009),Han,Yang and Xiao,Internat.J.Bifur.Chaos 22(2012),Atabaigi,Zangeneh and Kazemi,Nonlinear Anal.75(2012),Xiong,Internat.J.Bifur.Chaos 25(2015)],such that our results can manage limit cycle bifurcation near and inside homoclinic loop in smooth differential systems with a nilpotent saddle or a nilpotent cusp of arbitrary order,and contain all known results as special cases.The reader can further study the bifurcation of limit cycles near ho-moclinic loop in smooth differential systems with arbitrary order nilpotent saddle or cusp by Melnikov function on basis of our results.