The Number Of Limit Cycles Of A Class Polynomial Differential Systems

One of the main problems in the qualitative theory of the planar differential systems is to study the number and distribution of their limit cycles. This paper is devoted to finding the number of limit cycles of a class polynomial differential systems.In Chapter 1, we introduce the second part of Hilberts 16th problem and some research about the problem, and some preliminary knowledge(including dynamic system , limit cycle and so on). Then, the author briefly introduces the research work of this paper. The research work of this paper consists of three chapters. The maincontent is to find the number of limit cycles of the integrable system, which is called the unperturbed system, with the perturbation of some polynomials.In Chapter 2, considering the perturbed system under the low degree perturbations, the author studies the number of limit cycles bifurcating from the period annulus of the unperturbed system. One can prove that with the perturbation of degree 1 or 2, the system has at most 1 limit cycle; and with the perturbation of degree 3 or 4, the system has at most 4 limit cycles.In Chapter 3, considering the perturbed system with the perturbation of any degree n, the author mainly studies the coefficient formulas of the first order Melnikov function M(h) of the perturbed system. By estimating the number of zeros of M(h), we get the lower bound of the Hopf cyclicity of the perturbed system at the origin is at leastIn Chapter 4, considering the perturbed system with the perturbation of degree n≥5 above, by calculating and simplifying M(h), we get the number of limit cycles bifurcating from the period annulus of the unperturbed system. is at least Using the method of the complex analysis and the Argument Principle, we obtain an upper bound for the number of limit cycles isOne can get the innovation of this paper from two aspects. For the research method, by the calculation and induction for the number of zeros of M(h), we find the upper and lower bound of the number of limit cycles and give the corresponding proof. From the composition of this paper, for this system, we discuss the Hopf and global bifurcation with the perturbation of any degree.