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The paper deals with the problem of existence of a convergent "strong" normalform in the neighbourhood of an equilibrium, for a finite dimensional system ofdifferential equations with analytic and time-dependent non-linear term. Theproblem can be solved either under some non-resonance hypotheses on thespectrum 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 inwhich, despite under weaker non-resonance hypotheses, the nonlinearity isrequired to be asymptotically autonomous. The result is obtained as aconsequence of the existence of a strong normal form for a suitable class ofreal-analytic Hamiltonians with non-autonomous perturbations.

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