平面二次系统极限环及其稳定与分岔的计算  

Calculation of Limit Cycles and Their Stability and Bifurcations for Planar Quadratic Differential Systems

作　　者：黄赪彪[1] 邬华东[2] 

机构地区：[1]中山大学力学系,广东广州510275 [2]中山大学科技处,广东广州510275

出　　处：《中山大学学报（自然科学版）》2008年第2期28-31,共4页Acta Scientiarum Naturalium Universitatis Sunyatseni

基　　金：国家自然科学基金资助项目(10672193)

摘　　要：平面二次多项式微分系统极限环的数目、函数表达式、在相平面上的形状和位置,及其在参数平面上的分岔曲线等,对应用科学,例如非线性振动、生态学或生物学等领域有重要意义。将平面二次多项式微分系统极限环相图的x坐标假设为广义谐函数;用增量迭代法近似算出极限环的y坐标、频率、周期、稳定性指标,以及极限环关于参数分岔曲线的表达式,这将为解决著名的Hilbert第16问题(第二部分当n=2)提供一种定性和定量分析的途径。并给出绕奇点(0,0)具有三个极限环的例子。It is very helpful to determine the number, the function expression, their shapes and positions in the phase plane and the bifurcation curves in the parameter plane of the plane quadratic differentiation system in the fields of ecological, biological and applied sciences, e. g. nonlinear oscillations. The x coordinates of limit cycle phase portraits for planar quadratic polynomial differential systems are supposed as the generalized harmonic function. The approximate analytical expressions of y coordinates, frequency, periodic, stability index and bifurcation about the parameter are calculated by alternate method. The present will provide a way to solve the known as the Hilbert＇s problem 16 （second part as n = 2 ）. Am example with three limit cycles surrounding the singular point （0, O） is shown.

关 键 词：平面二次多项式微分系统 极限环 频率 稳定性 分岔 

分 类 号：O322[理学—一般力学与力学基础]