Global Dynamics Of Three Classes Of Liénard Systems

The study of Liénard system plays an important role not only in differential equation theory but also in dynamical system,which can be widely applied to physical,mechanical,biological models and so on.In this thesis,we study the global dynamics of some Liénard systems.A non-symmetric Liénard system,a Liénard system with(Z₂-symmetry and a quintic Liénard system are mainly discussed.The whole thesis consists of five chapters.In Chapter 1,we introduce the significance of studying differential equations.Addition-ally,the historical background,the latest research progress of the global dynamics of Liénard system are introduced.Meanwhile,the research methods and main content of this paper is briefly concluded.In Chapter 2,we study the global dynamics of a non-symmetric Liénard system.Firstly,we consider the properties of equilibria at finity and infinity.Then,we analyze the existence of limit cycles,homoclinic orbits and the heteroclinic orbits by the indices of equilibria and the change of manifold,and then we obtain the global dynamics of this system.In Chapter 3,we mainly study the global dynamics of a Liénard system with Z₂-symmetry,and analyze the case that the center is nilpotent center.Firstly,we consider the local dynamics of this system.We analyze the qualitative properties of all equilibria at finity and infinity.The bifurcation of elementary equilibria and nilpotent equilibria in small neighborhood is analyzed and the local dynamics is explored.Then according to the criterions,we analyze the nonexistence of limit cycles and the number of limit cycles in some parameter regions,also we prove that this system has at most two limit cycles.We get that the manifold of the saddle point will change with the change of the parameters by calculation.Finally,combining the annular region theorem,the stability of equilibria,the change of manifolds and symmetry,the dynamical behaviour of this system with nonlocal parameters is analyzed,and then we obtain the complete global dynamics of this system.In Chapter 4,we study the global dynamics of a quintic Liénard system with asymmetric vector field.Firstly,we analyze the qualitative properties of equilibria.The topological classification and stability of equilibria in different regions of parameter are judged.Then,we prove that the nonexistence of limit cycles and this system has at most one limit cycle by the criterions respectively.Then through calculation,we analyze the change of stable and unstable manifold of the saddle points.Finally,using comparison theorem,the change of manifold and annular region theorem,it is concluded that this system will occur Hopf bifurcation,homoclinic bifurcation,two-saddle heteroclinic bifurcation and so on,we also give the relative positions of bifurcation curves and the dynamical behavior of this system with nonlocal parameters is obtained.In Chapter 5,the main work of the thesis is summarized and some prospects for future research work are put forward.