具有中心奇点的非线性方程相轨线的全局结构

Global Structure of the Phase Orbit of Nonlinear Differential Equations with a Center

线性微分方程的奇点是中心,附加非线性项以后,奇点可能仍然为中心,也可能改变拓扑结构变为焦点.本文讨论这类非线性微分方程在只有一个奇点的情况下轨线的全局结构,提出在大范围内轨线的拓扑结构可能是:①围绕奇点的一簇封闭曲线;②围绕奇点的螺旋线;③存在一条或两条分界线,在分界线里面是一簇围绕奇点的封闭曲线,在分界线外面是逸散的曲线;④没有极限环.

If the singular point of linear differentiaI equations is the center, after adding nonlinear terms the singular point may still be the center or become a focus owing to the change of the topological structure. This paper discusses the global structure of the orbit in the phase plane of the nonlinear differential equation with only one singular point. We show that in the global fieds the topological structure of the phase orbit may be one of the following cases: 1) It is a bunch of the closed orbits around the singular point. 2) It is the spiral curve around the singular point. 3) There exist one or two demarcation lines. It is a bunch of the closed orbits around the singular point inside tho demarcation line and the dispersed orbit outside the demarcation line. 4) There is no limit cycle.