Convergence versus integrability in Poincaré-Dulac normal form

We show that, to find a Poincaré-Dulac normalization for a vector field is the same as to find and linearize a torus action which preserves the vector field. Using this toric characterization and other geometrical arguments, we prove that any local analytic vector field which is integrable in the non-Hamiltonian sense admits a local analytic Poincaré-Dulac normalization. These results generalize the main results of our previous paper \cite{ZungBirkhoff} from the Hamiltonian case to the non-Hamiltonian case. Similar results are presented for the case of isochore vector fields.

Full Text (PDF format)

Published 1 January 2002