### Abstract

In the study of singularities of vector fields on the plane the analysis of perturbations of Hamiltonian systems is crucial. The number of isolated limit cycles in the perturbed system is related to the number of zeros of periods (Abelian integrals). If the Hamiltonian function is algebraic, then the there are finitely many independent periods. They satisfy a matrix linear homogeneous differential equation, the Picard-Fuchs equation. See for instance [BK81]. The average of the perturbed vector field over the periodic orbits of the unperturbed problem can be expressed in those periods.

Original language | English |
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Title of host publication | Dynamics, Bifurcation and Symmetry |

Subtitle of host publication | New Trends and New Tools |

Editors | Pascal Chossat |

Place of Publication | Dordrecht |

Publisher | Kluwer |

Pages | 333-341 |

Number of pages | 10 |

ISBN (Electronic) | 978-94-011-0956-7 |

ISBN (Print) | 978-94-010-4413-4 |

DOIs | |

Publication status | Published - 1994 |

Event | NATO Advanced Research Workshop on Dynamics, Bifurcation and Symmetry 1993: New Trends and New Tools - Cargèse, France Duration: 3 Sep 1993 → 9 Sep 1993 |

### Publication series

Name | NATO ASI Series C |
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Publisher | Kluwer Academic |

Volume | 437 |

ISSN (Print) | 0377-2071 |

### Conference

Conference | NATO Advanced Research Workshop on Dynamics, Bifurcation and Symmetry 1993 |
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Country | France |

City | Cargèse |

Period | 3/09/93 → 9/09/93 |

Other | E.B.T.G. Conference |

### Keywords

- Hamiltonian function
- Abelian integral
- Unperturbed problem
- Monotonicity result
- Homoclinic connection

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## Cite this

van Gils, S. A. (1994). An inhomogeneous Picard-Fuchs equation. In P. Chossat (Ed.),

*Dynamics, Bifurcation and Symmetry: New Trends and New Tools*(pp. 333-341). (NATO ASI Series C; Vol. 437). Dordrecht: Kluwer. https://doi.org/10.1007/978-94-011-0956-7_27