非线性微分系统的极限环分支研究

Qualitative theory of differential equations and bifurcation theory of dynamical systems are important research areas of modern mathematics. There are not only some classical problems, which remain unsolved for more than one hundred years, but also some important questions which arise from the real world. During the last several decades, high-dimensional systems and non-smooth systems have received a great deal of attentions at home and abroad. This project mainly studies the number of limit cycles in planar systems, bifurcations of algebraic limit cycles and bifurcations of periodic orbits in high-dimensional systems. Our research objects include smooth systems and non-smooth systems. About our research method, we attach great attention to the combination of theoretical analysis with symbolic computation. Our main purpose is to have some innovations in research theories and methods, like obtaining the number of limit cycles for some planar polynomial systems and piecewise-smooth systems, and some new existence conditions of some algebraic limit cycles, and new methods for the study of bifurcations in high-dimensional systems under perturbations.

微分方程定性理论和动力系统分支理论是现代数学的一个重要研究领域，这里既有百多年未解的经典问题，也有源于实际问题的热门课题。近几十年来，高维系统和非光滑系统由于其越来越广泛的应用背景，吸引了众多国内外学者的注意力。本项目主要研究平面系统中的极限环个数、代数极限环分支、以及高维系统中的周期解分支，研究对象包括光滑系统和非光滑系统，在研究方法上我们将更注重理论分析与符号计算相结合，以期能够在研究理论和方法上有所创新，获得一些平面多项式系统或者分段光滑系统的极限环个数、几类代数极限环存在的新条件，以及高维系统扰动分支中新的研究方法。

本项目主要研究了光滑系统和分片光滑系统的极限环分支。通过符号计算得到焦点量或Lyapunov常数，依据其相互独立性得出了几类微分系统Hopf分支极限环的最大个数或者新的个数估计，并为Melnikov函数在异宿环周围展开式计算提出了新的迭代算法。