Stability And Bifurcation Analysis Of Two Kinds Of Dynamical Systems

Dynamical system is the important component of nonlinear science and is one of important branches of modern mathematics. Chaos and bifurcation is the new forms of nonlinear dynamical systems and is an important research area. The research achievements can be widely used in physics、mechanics、chemistry、biology and economics, etc, which has great practicability. In this thesis, the dynamic characteristics of a jerky dynamic and an SEIV epidemic model are discussed as the following four chapters.In the first chapter, the research background and the current situation are introduced. In the second chapter, the necessary knowledge of discussing the dynamic properties is in-troduced. In the third chapter, the chaos and Hopf bifurcation analysis of a jerky dynamic is considered. The stability of equilibria and the existence of Hopf bifurcation are ana-lyzed. By using the normal form theory, the direction and stability of bifurcating periodic solutions are investigated. Finally, some numerical examples are performed which are con-sistent with the theory. And we find that the periodic-doubling sequence of bifurcations lead to a Feigenbaum-like attractor. In the fourth chapter, an SEIV epidemic model for childhood disease with partial permanent immunity is studied. The basic reproduction number is worked out. And we have analyzed the local and global asymptotical stability of the equilibria respectively. The bifurcation phenomenon at the equilibrium is analyzed and the results show that periodic orbits will bifurcate from the endemic equilibrium when the treated rate τ crosses a critical value. Finally, some numerical examples are performed vhich are consistent with the theory.