Study On Bifurcation And Chaos Of Two Nonlinear Dynamical Systems

In this thesis, we discuss the complex dynamics of two classes of nonlinear systems as the bifurcation parameters are changed by using bifurcation and chaos theories of dynamical systems. The systems exhibit the static and dynamical bifurcations as well as chaos, including saddle-node bifurcation, Hopf bifurcation, BT bifurcation and so on. In particular, the BT bifurcation is analyzed by use of a global qualitative method, and the curves of saddle-node bifurcation, Hopf bifurcation and homoclinic bifurcation are obtained at the degenerate equilibrium. Furthermore, the potential chaotic motions are further studied by means of Melnikov's method and the parameters for the occurrence of chaotic motions are obtained.The thesis consists of four chapters as follows:In Chapter 0 and 1, the brief review of bifurcation theory and chaos theory and their development are presented, respectively. The basic conceptions and research methods are introduced and the necessary and sufficient conditions of saddle-node bifurcation, transcritical bifurcation, pitchfork bifurcation and Hopf bifurcation are given. At the end of Chapter 0, the arrangement of this thesis is shown briefly.In Chapter 2, the nonlinear dynamical model of ion channel in cell membrane is taken for as a study object, the existence and stability of its steady states are qualitatively analyzed. The degenerate critical point, the saddle-node bifurcation and Hopf bifurcation are discussed. Moreover, its potential chaotic motion is further studied by means of Melnikpv's method and the parameters for the occurrence of chaotic motion are obtained. So it is tried to prove that the aberrance occurred in the dynamic biological process may be accounted for some diseases, and offered the mechanism explain and the clue of the medicament study.In Chapter 3, the existence and stability of the disease-free equilibrium and the endemic equilibrium for the SIS epidemic model with nonlinear incidence rate are qualitatively analyzed, considering the factor of population dynamics such as the disease-related and the natural mortality and the constant recruitment of population. The existence and stability of the limit cycle is also discussed by using Dulac function and Hopf bifurcation theory. Finally, the BT bifurcation is analyzed by using a global qualitative method and the curves of saddle-node bifurcation, Hopf bifurcation and homoclinic bifurcation are obtained at the degenerate equilibrium.