一类拟三次系统的中心、极限环分支及等时中心

【作者】 赵百利；

【导师】 刘一戎；

【作者基本信息】 中南大学 ， 应用数学， 2007， 硕士

【摘要】 本文主要研究拟解析系统的中心条件、极限环分支及等时中心问题,全文共由三章组成。第一章对平面多项式微分系统的中心-焦点判定、极限环分支及等时中心问题的历史背景及研究现状进行了概述。第二章研究了一类拟三次系统的中心条件与极限环分支问题。首先通过适当的变换将拟解析系统的原点(或无穷远点)转化为解析系统的原点,推导出了计算原点奇点量的线性递推公式,然后在个人计算机上用Mathematica软件计算出该系统原点的前18个奇点量,从而导出原点成为中心的条件与15阶、18阶细焦点的条件,在此基础上用两种不同的方法,在不构造poincare环域的情况下,给出了拟三系统在原点可以扰动出5个小振幅的极限环的一个实例(本文内容发表在《黑龙江科技学院学报》2007,17(3)上),并证明其中有3个是稳定的。第三章研究了第二章中的一类拟三次系统的等时中心。首先用一种新方法求出了计算系统原点周期常数的线性递推公式,然后用Mathematica软件求出该系统原点的前9个周期常数,从而得到了中心成为等时中心的必要条件,并利用一些有效途径证明了这些条件的充分性。更多还原

【Abstract】 This thesis is devoted to the problems of the center conditions, bifurcation of limit cycles and the isochronous conditions for quasi analytic systems. It is composed of three chapters.In chapter 1, the historical background and the present progress of problem about center conditions, bifurcation of limit cycles and isochronous conditions of planar polynomial differential system are introduced and summarized.In chapter 2, center condition and bifurcation of limit cycles of quasi cubic system are investigated. By converting the origin or the singular point at infinity of quasi analytic system into the origin of a equivalent complex analytic system, the recursion formula for computation of singular point quantities are given, and, with computer algebra system Mathematica, the first 18 singular point quantities are deduced. At the same time, the conditions for the origin to be a center and 15-order and 18-order fine focus are derived respectively. A quasi cubic system that bifurcated 5 limit cycles, three of which are stable, at the origin is obtained. This result was published on《Journal of Heilongjiang Institute of Science&Technology》2007,17(3).In chapter 3, the problem of the isochronous conditions of quasi cubic system in chapter 2 is investigated. Firstly, by using a new method, the recursion formula for calculating periodic constants are given, then the first 9 periodic constants are deduced. At the same time, necessary conditions for center to be an isochronous center are given, and proof of isochronous these systems by using some effective methods are given, too.更多还原