The Qualitative Analysis Of Some Polynomial Systems

In this paper, we consider the limit cycle of several kinds of higher degree polynomial systems and theω-limit sets of dynamical systems.By using the qualitative analysis theory of differential equations, we obtain the conditions of the nonexistence of the limit cycles of higher degree polynomial systems. Then we transform the higher degree polynomial system into the generalized Liénard system and discuss the equilibrium and the limit cycle for this system by using the abundant results of the generalized Liénard system, we analyze the type of the equilibrium and establish the conditions of the existence and the instability of the limit cycles.Conley theory has been successfully applied in the study of the qualitative analysis of the dynamical systems. In this paper, we point out that the definition of theω-limit set given by Conley is different from that defined by N.P.Bhatia. We discuss theω-limit sets of dynamical systems using the Conley theory, chain recurrence and Morse decomposition theory.The layout of this paper is as follows: in Chapter 1, an introduction and main methods using in this paper are presented. The concepts and theories of dynamical systems are given in Chapter 2. In Chapter 3, the qualitative properties of several kinds of higher degree polynomial systems are discussed, the conditions of the existence and nonexistence of the limit cycle for these systems are obtained, some results of Refs. [1, 2] are generalized. In Chapter 4, theω-limit sets of dynamical systems are studied by using the Conley theory, some new results are given, and the related results of J.Schropp are generalized. Finally, the problem for further studies is proposed in Chapter 5.