等变多项式系统的全局与局部分支

Our research results are made of two parts. The first part is the research on bifurcation of polynomial systems. The second part is the research on traveling wave solutions of partial differential equations. In the frest part, we find that five perturbed systems have the same distribution of limit cycles. We show that there are 13 limit cycles for cubic systems with 7 degrees perturbations, 14 limit cycles for cubic systems with 9 degrees perturbations. We also give qualitative analysis method and numerical.exploration to the study of limit cycles for quitic systems. In the second.part, by using bifurcation methods of dynamical systems, we investigate.traveling wave solutions of several famous partial differential equations.The explicit expressions of soliton solutions and kink solutions are given. Our works extend some results in literary. Specially, we obtain some new soliton solutions of KdV equation and Camassa-Holm equation.

本项目拟研究三次以上对称多项式系统的全局与局部分支。着重研究在对称群作用下不变的哈密顿向量场及其扰动系统的小振幅周期解、同宿环、异宿环的存在性，极限环的分布与个数。寻找适合于高次多项式系统的研究方法，编制或改进计算软件包，确定计算机模拟的方法，深入探讨与弱化的希尔伯特第16问题有关的平面多项式系统的动力学性质。