关于相位分布的一个注记

A remark on configurations

讨论微分方程定性理论中长时间未解决的一个极限集的相位分布图的不可能性 .在证明中 ,作者定义的向量场同胚映射起着举足轻重的作用 .

A long standing open problem in qualitative analysis of differential equations stated by Andronov and Khaiken is to show the possibility of the appearance of a limit cycle arising out of a separatrix.This question is of great interest from the point of view of the theory of differential equations and physics in general,and its analysis however presents certain difficulties which have not yet been overcome.Numerical solutions have shown the possibility of the appearance of different configurations.This paper indicates that numerical results and theoretical results have a difference because of limit sets which is a critical fact of chaotic motions in computing.For any planar analytic antonomous system,the corresponding vector field homeomorphism is applied to show the impossibility of the appearance of a limit cycle arising out of separatrix.Topological proof may lead to a different conclusion from mathematical analysis.A characteristic of indices of limit sets (the type Ⅰ and the type Ⅱ)given by the author in another paper also disproves the possibility of the appearance of a limit cycle arising out of a separatrix.The achievement in the paper can be extended to differentiable systems.As a typical system the Duffing equation is considered.