Integrability and strong normal forms for non-autonomous systems in a neighbourhood of an equilibrium

TL;DR

This paper establishes conditions under which a convergent strong normal form exists near an equilibrium for finite-dimensional, time-dependent nonlinear systems, extending previous work by Pustil'nikov.

Contribution

It completes and extends Pustil'nikov's work by providing conditions for strong normal forms with weaker non-resonance hypotheses and decaying nonlinearities.

Findings

Existence of convergent strong normal forms under non-resonance conditions.

Extension of Pustil'nikov's results to broader classes of non-autonomous systems.

Application to real-analytic Hamiltonians with time-dependent perturbations.

Abstract

The paper deals with the problem of existence of a convergent "strong" normal form in the neighbourhood of an equilibrium, for a finite dimensional system of differential equations with analytic and time-dependent non-linear term. The problem can be solved either under some non-resonance hypotheses on the spectrum of the linear part or if the non-linear term is assumed to be (slowly) decaying in time. This paper "completes" a pioneering work of Pustil'nikov in which, despite under weaker non-resonance hypotheses, the nonlinearity is required to be asymptotically autonomous. The result is obtained as a consequence of the existence of a strong normal form for a suitable class of real-analytic Hamiltonians with non-autonomous perturbations.