Qualitative Analysis Of Several Categories Higher Polynomial Systems

In this paper, by using the qualitative theories and bifurcation method of ordinary differential equations, several categories higher polynomial systems with universal significance are researched, qualitative behaviors of trajectories and some conclusion with universal significance are obtained. The whole paper consists of three chapters.The first chapter is the introduction, in which we introduce the developing history of limit cycle and bifurcation theories, and the present progress of higher polynomial systems. Finally, some fundamental definitions and conclusions of dynamics systems such as bifurcation and stability theory that can be used in this paper were given, and briefly represent the main works of the thesis.In the second chapter, a class of higher polynomial system with universal significance is researched with the theory of limit cycles and bifurcation of the plane autonomous system. The system iswhere n>0,p≥0,αis a plus odd number, h(y) is an arbitrary polynomial functionis whose degree is greater than two and meet the need of h(0) = 0. Inthis chapter, the location of the limit cycles are analyzed importantly, at the same time, the problems of existence and uniqueness and stability of the limit cycles are solved.In the third chapter, a class of codimension-two higher planar polynomialsystem (?) is researched, where P_n,Q_n are polynomials whose degree are greater than 2. The universal unfoldings is(n is odd number)and(n is even number)whereμ₁,μ₂∈R.In this chapter, method of the center manifold, theorems ofbifurcation, theorem of rotary vector field are used to discuss the two situations and the bifurcations are solved when the parameters meet the needs of some different conditions, finally, the full orbit bifurcation figure is given.