Mathematical Research Letters

Volume 9 (2002)

Number 2

Convergence versus integrability in Poincaré-Dulac normal form

Pages: 217 – 228



Nguyen Tien Zung (Université Montpellier II)


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.

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