High-Order Melnikov Method for Time-Periodic Equations

Fengjuan Chen, Qiudong Wang

Research output: Contribution to journalArticlepeer-review

7 Scopus citations


This paper discusses a high-orderMelnikov method for periodically perturbed equations.We introduce a new method to computeMk(t0) for all k 0, amongwhichM0(t0) is the traditionalMelnikov function, and M1(t0), M2(t0), . . . are its high-order correspondences. We prove that, for all k 0, Mk(t0) is a sum of certain multiple integrals, the integrand of which we can explicitly compute. In particular, we obtain explicit integral formulas for M0(t0) and M1(t0). We also study a concrete equation for which the explicit formula of M1(t0) is used to prove the existence of a transversal homoclinic intersection in the case of M0(t0) 0.

Original languageEnglish (US)
Pages (from-to)793-818
Number of pages26
JournalAdvanced Nonlinear Studies
Issue number4
StatePublished - Oct 1 2017


  • High-Order Melnikov Method
  • Time-Periodic Equation
  • Transversal Homoclinic Intersection

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • General Mathematics


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