## 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) |
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Pages (from-to) | 1095-1128 |

Number of pages | 34 |

Journal | Journal of Differential Equations |

Volume | 267 |

Issue number | 2 |

DOIs | |

State | Published - Jul 5 2019 |

## Keywords

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

## ASJC Scopus subject areas

- Analysis
- Applied Mathematics