Convergence versus integrability in Poincaré-Dulac normal form

Abstract

Abstract. 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 nonHamiltonian sense admits a local convergent Poincaré-Dulac normalization. These results generalize the main results of our previous paper [12] from the Hamiltonian case to the non-Hamiltonian case. Similar results are presented for the case of isochore vector fields.

Extension and Unification of Singular Perturbation Methods for ODEs Based on the Renormalization Group Method

“…It is proved that the infinite order RG equation converges if and only if the original equation is invariant under an appropriate torus action (Thm.5.1). This result extends Zung's theorem [54] which gives a necessary and sufficient condition for the convergence of normal forms of infinite order. The infinite RG equation for a time-dependent linear system proves to be convergent (Thm.5.6) and be related to monodromy matrices in Floquet theory.…”

“…is analytic with respect to x ∈ C n and ε ∈ I ⊂ R. 1 , · · · , ib n ), where i = √ −1 and b j ∈ Z for j = 1, · · · , n (see Murdock [39], Zung [54]). Let p be the maximum number of linearly independent such matrices B 1 , · · · , B p and call it the toric degree of F. Then, the matrix e…”

Normal Forms of Maps: Formal and Algebraic Aspects

“…With regard to simultaneous linearizations of commuting holomorphic maps of (C n , 0), n 2 we refer to [12], where the linear parts are assumed semisimple. For resonant normal forms of commuting holomorphic vector fields under a simultaneous version of Bruno's Condition ω, we refer to Stolovich [21] and the references therein (see also Yoshino [24], Zung [25], where the presence of the Jordan blocks is allowed in some cases).…”

Convergent and Divergent Solutions of Singular Partial Differential Equations with Resonance or Small Denominators

“…(cf. [1] and [5]. ) We shall give a rather simple wide class of nonlinear perturbations for which one can always find a convergent solution, which is different from an integrability condition because we put no restriction on the resonance dimension.…”

“…(cf. [1] and [5].) We also construct a Liouville type linear part and a nonlinear perturbation for which a divergence of a (unique) solution occurs.…”