### 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 |

### Keywords

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

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics