High order Melnikov method

Theory and application

Fengjuan Chen, Qiu-Dong Wang

Research output: Contribution to journal › Article

Abstract

Let D(t 0 ,ε) be the splitting distance of the stable and unstable manifold of a time-periodic second order equation. We expand D(t 0 ,ε) as a formal power series in ε as D(t 0 ,ε)=E 0 (t 0 )+εE 1 (t 0 )+⋯+ε n E n (t 0 )+⋯. In this paper we derive an explicit integral formula for E 1 (t 0 ). We also evaluate E 1 (t 0 ) to prove the existence of homoclinic tangles for an equation to which the Poincaré/Melnikov method fails to apply.

Original languageEnglish (US)
JournalJournal of Differential Equations
DOIs
StatePublished - Jan 1 2019

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Melnikov Method
Stable and Unstable Manifolds
Tangles
Formal Power Series
High-order Methods
Integral Formula
Homoclinic
Second Order Equations
Expand
Explicit Formula
Evaluate

Keywords

  • High order Melnikov method
  • Homoclinic intersection
  • Time periodic equation

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

Cite this

High order Melnikov method : Theory and application. / Chen, Fengjuan; Wang, Qiu-Dong.

In: Journal of Differential Equations, 01.01.2019.

Research output: Contribution to journal › Article

Chen, F & Wang, Q-D 2019, 'High order Melnikov method: Theory and application', Journal of Differential Equations. https://doi.org/10.1016/j.jde.2019.02.003
Chen F, Wang Q-D. High order Melnikov method: Theory and application. Journal of Differential Equations. 2019 Jan 1. https://doi.org/10.1016/j.jde.2019.02.003
Chen, Fengjuan ; Wang, Qiu-Dong. / High order Melnikov method : Theory and application. In: Journal of Differential Equations. 2019.
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AB - Let D(t 0 ,ε) be the splitting distance of the stable and unstable manifold of a time-periodic second order equation. We expand D(t 0 ,ε) as a formal power series in ε as D(t 0 ,ε)=E 0 (t 0 )+εE 1 (t 0 )+⋯+ε n E n (t 0 )+⋯. In this paper we derive an explicit integral formula for E 1 (t 0 ). We also evaluate E 1 (t 0 ) to prove the existence of homoclinic tangles for an equation to which the Poincaré/Melnikov method fails to apply.

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