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The paper deals with the problem of the existence of a normal form for anearly-integrable real-analytic Hamiltonian with aperiodically time-dependentperturbation decaying (slowly) in time. In particular, in the case of anisochronous integrable part, the system can be cast in an exact normal form,regardless of the properties of the frequency vector. The general case istreated by a suitable adaptation of the finite order normalization techniquesusually used for Nekhoroshev arguments. The key point is that the so called"geometric part" is not necessary in this case. As a consequence, no hypotheseson the integrable part are required, apart from analyticity. The work, based ontwo different perturbative approaches developed by A.Giorgilli et al., is ageneralisation of the techniques used by the same authors to treat morespecific aperiodically time-dependent problems.

Negligibility of small divisor effects in the normal form theory for nearly-integrable Hamiltonians with decaying non-autonomous perturbations