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

Paul, E.

J. Dyn. Control Syst. 10, No. 4, 545-575 (2004).

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

Keywords:

Singularities; normal forms; Hamiltonian vector fields; relative cohomology; foliations