# High order Melnikov method: Theory and application

Fengjuan Chen, Qiu-Dong Wang

Research output: Contribution to journalArticle

### 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 language English (US) Journal of Differential Equations https://doi.org/10.1016/j.jde.2019.02.003 Published - Jan 1 2019

### Fingerprint

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 journalArticle

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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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