Study On The Dynamic Properties Of Several Classes Of Three-Dimensional Differential Dystems

In this thesis,by qualitative theory and bifurcation theory of differential equations,we study zero-Hopf bifurcation,multiple Hopf bifurcation,Pitchfork bifurcation,dynamical properties at infinity,and limit cycle problems for several types of differential systems,which are composed of five chapters.In Chapter 1,we introduce the research background and the latest research status of the problems for bifurcation and chaos,limit cycles.And summarizes the content and innovation of each chapter.In Chapter 2,the zero-Hopf bifurcation problem of a class of generalized Lorenz chaotic systems near finite and infinite singular points is studied.By using the average method and the normal form,we prove that at most one limit cycle can bifurcate near the finite equilibrium point.By using the average method and the normal form,we prove that at most one limit cycle can be obtained near the zero-Hopf equilibrium point in a finite plane.The dynamic behavior and zero-Hopf bifurcation of the system at infinity are analyzed,and the uniqueness of the limit cycle at the equilibrium point at infinity is obtained.The global structure on the Poincaré disk is also given.In Chapter 3,the multiple bifurcation of limit cycles for a segmented disc dynamo system is studied.For two cases of the segmented disc dynamo system,namely the system withor without friction coefficient(abbr.SDDF-or SDD-model),the maximum number of limit cycles is obtained at the symmetrical equilibrium points under the condition of synchronous perturbation respectively.And the parameters condition is classified for exact number of limit cycles near each weak focus.In Chapter 4,the dynamic properties of a class of tumor models are studied,and the existence and stability of all positive equilibrium points are described in detail.Firstly,the parameter interval of the Transcritical bifurcation and a Pitchfork bifurcation at the single cell equilibrium point is determined.Secondly,the necessary and sufficient conditions for bifurcation of three limit cycles at the dual cell equilibrium point is obtained,which explains the bistability phenomenon between immune cells and tumor cells.Finally,the parametric conditions for the zero-Hopf bifurcation at the three cell equilibrium point to produce a stable limit cycle are analyzed.