Formal normal forms for the perturbations of a quasi-homogeneous Hamiltonian vector field.

*(English)*Zbl 1068.37032Summary: We classify up to a formal change of the variables the vector fields of two complex variables together with foliations that they define which are a perturbation of a quasihomogeneous Hamiltonian vector field \(X_0\) by terms of higher degree of quasihomogeneity. We do not require anything about the degree \(\delta_0\) of the initial vector field \(X_0\), but we assume that the perturbed vector field still keeps invariant the separatrices of \(X_0\). We obtain formal normal forms which extend those obtained in the case of an initial vector field with a semisimple or nilpotent linear part. We give an interpretation of the dual version of these normal forms through the relative cohomology with respect to the dual initial differential form \(\omega_0\).

##### MSC:

37F75 | Dynamical aspects of holomorphic foliations and vector fields |

37G05 | Normal forms for dynamical systems |

34C20 | Transformation and reduction of ordinary differential equations and systems, normal forms |

34M35 | Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms |

32S65 | Singularities of holomorphic vector fields and foliations |

37C10 | Dynamics induced by flows and semiflows |

37C15 | Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems |