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Formal normal forms for the perturbations of a quasi-homogeneous Hamiltonian vector field. (English) Zbl 1068.37032
Summary: 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
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