High order Melnikov method: Theory and application

Fengjuan Chen; Qiudong Wang

doi:10.1016/j.jde.2019.02.003

High order Melnikov method: Theory and application

Fengjuan Chen, Qiudong Wang

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

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)
Pages (from-to)	1095-1128
Number of pages	34
Journal	Journal of Differential Equations
Volume	267
Issue number	2
DOIs	https://doi.org/10.1016/j.jde.2019.02.003
State	Published - Jul 5 2019

Keywords

High order Melnikov method
Homoclinic intersection
Time periodic equation

ASJC Scopus subject areas

Analysis
Applied Mathematics

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10.1016/j.jde.2019.02.003

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title = "High order Melnikov method: Theory and application",

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 Poincar{\'e}/Melnikov method fails to apply.",

keywords = "High order Melnikov method, Homoclinic intersection, Time periodic equation",

author = "Fengjuan Chen and Qiudong Wang",

note = "Funding Information: Research supported by National Nature Science Foundation of China (Nos. 11171309, 11471289). Publisher Copyright: {\textcopyright} 2019 Elsevier Inc.",

year = "2019",

month = jul,

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doi = "10.1016/j.jde.2019.02.003",

language = "English (US)",

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TY - JOUR

T1 - High order Melnikov method

T2 - Theory and application

AU - Chen, Fengjuan

AU - Wang, Qiudong

PY - 2019/7/5

Y1 - 2019/7/5

N2 - 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.

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.

KW - High order Melnikov method

KW - Homoclinic intersection

KW - Time periodic equation

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DO - 10.1016/j.jde.2019.02.003

M3 - Article

AN - SCOPUS:85061381971

SN - 0022-0396

VL - 267

SP - 1095

EP - 1128

JO - Journal of Differential Equations

JF - Journal of Differential Equations

IS - 2

ER -

High order Melnikov method: Theory and application

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Keywords

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