超临界情形的Melnikov函数及应用

数学物理学报 ›› 1997, Vol. 17 ›› Issue (2): 152-162.

超临界情形的Melnikov函数及应用

张建文, 李庆士

太原工业大学数学系太原 030024

收稿日期:1995-03-31 修回日期:1995-08-08 出版日期:1997-04-26 发布日期:1997-04-26

Melnikov Function and Application in Supercritical Cases

Zhang Jianwen, Li Qingshi

Taiyuan University of Technology, Taiyuan 030024

Received:1995-03-31 Revised:1995-08-08 Online:1997-04-26 Published:1997-04-26

摘要/Abstract

摘要： 该文具体推导了三阶Melnikov函数的积分表达,解决了电机工程中提出的一类系统(见[5]),当参数μ=(2)/21时的超临界(一阶、二阶Melnikov函数恒为零)的情形下,系统的稳定流形与不稳定流形的相对位置的确定问题.并通过环面上的VanderPol方程,对[2]与[4]所给的二阶Melnikov函数的表达式进行了比较,结果发现[2]所给的平面自治系统的二阶,n阶表达式均是错的.该文在最后作了纠正.

关键词: 双曲鞍点, 稳定流形, 不稳定流形, 同宿轨, 异宿轨

Abstract: This paper deduces the formula of the third order Melnikov function. the relativc positions of the stable manifold and unstable manifold of the systems in electrical engineering are determined in supercritical cases, We make a comparison between the formulae of thesecond order Melnikov functions in[2] and[4] from studying of toral van der Pol equations)and find that the second order and the nth order formulae in[2] are wrong. Finaly,we correct this mistake.

Key words: hyperbolic saddle point, stable manifold, unstable manifold, homoclinic orbits, heleroclinic orbits

张建文, 李庆士. 超临界情形的Melnikov函数及应用[J]. 数学物理学报, 1997, 17(2): 152-162.

Zhang Jianwen, Li Qingshi. Melnikov Function and Application in Supercritical Cases[J]. Acta mathematica scientia,Series A, 1997, 17(2): 152-162.

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