### 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 | Journal of Differential Equations |

DOIs | |

State | Published - Jan 1 2019 |

### Fingerprint

### 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.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - High order Melnikov method

T2 - Theory and application

AU - Chen, Fengjuan

AU - Wang, Qiu-Dong

PY - 2019/1/1

Y1 - 2019/1/1

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

UR - http://www.scopus.com/inward/record.url?scp=85061381971&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85061381971&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2019.02.003

DO - 10.1016/j.jde.2019.02.003

M3 - Article

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

ER -