Equivalence of the Melnikov Function Method and the Averaging Method

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• 作者：Maoan Han ; Valery G. Romanovski ; Xiang Zhang
• 关键词：Averaging method ; Melnikov function ; Limit cycle bifurcation
• 刊名：Qualitative Theory of Dynamical Systems
• 出版年：2016
• 出版时间：October 2016
• 年：2016
• 卷：15
• 期：2
• 页码：471-479
• 全文大小：426 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Dynamical Systems and Ergodic Theory
  Difference and Functional Equations
  
• 出版者：Birkh盲user Basel
• ISSN：1662-3592
• 卷排序：15

文摘

There is a folklore about the equivalence between the Melnikov method and the averaging method for studying the number of limit cycles, which are bifurcated from the period annulus of planar analytic differential systems. But there is not a published proof. In this short paper, we prove that for any positive integer k, the kth Melnikov function and the kth averaging function, modulo both Melnikov and averaging functions of order less than k, produce the same number of limit cycles of planar analytic (or \(C^\infty \)) near-Hamiltonian systems.