关于常微分方程区域分析理论中的极限环问题

来源期刊：中南大学学报(自然科学版)1985年第3期

论文作者：刘一戎

文章页码：116 - 125

关键词：极限环； 常微分方程； 分析理论； 全局渐近稳定； 区域； 简单闭曲线； 奇点； 环域定理； 闭轨； 分析方法

摘    要：本文举出反例说明文献[1]中提出的研究极限环问题的区域分析方法在理论上不具一般性。然后,建立了极限环存在的一个充分必要条件,并举例说明了所给的方法对于研究极限环存在与否的问题以及全局渐近稳定的问题是有效的。

Abstract: The method of regional analysis is advanced in [1] by Professor Qnin Yuanxun. it is a very convenient and effective method used to research into the quatitative problem of the system (E) x=X (x, y) y=Y (x, y) (x, Y∈C’) but it has not been proved on maths in [1]. In this paper, we present some opposed examples on the study of limit-cycle problem, of (E). We find that this method isn’t of general quality in theory. And, we set up a method to research into the problem on limit-cycle of (E). Let R(L) denote a region enclosed by a curve L, and D(L₂, L₁)== def R(L₂) - R(L₁), where R(L₁)∈R(L₂). When we have known that there exists a closed curve L within D(L₂, L₁), which is a Bendixson-curve or limit- cycle, and that no critical point lies D (L₂, L₁), we can use the theorem as follows to research into the limit-cycle problem. [Theorem] i) If I(L₂, L₁)≤0, then all positive half-path meeting L will not cross out of L, as t increases ii) if J(L₂, L₁)≥θ, then all positive half-path meeting L will not come into its interior, as t increases; iii) If L is a limit-cycle, then I (L₂, L₁)≥0≥J (L₂, L₁) where I(L₂, L₁)=∫∫Pdxdy+1/2∫∫[P+|P|]dxdy I(L₂, L₁)=∫∫Pdxdy+1/2∫∫[P-|P|]dxdy P=бX/бx+бY/бy.