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We consider one-dimensional systems in the presence of a quasi-periodicperturbation, in the analytical setting, and study the problem of existence ofquasi-periodic solutions which are resonant with the frequency vector of theperturbation. We assume that the unperturbed system is locally integrable andanisochronous, and that the frequency vector of the perturbation satisfies theBryuno condition. Existence of resonant solutions is related to the zeroes of asuitable function, called the Melnikov function - by analogy with the periodiccase. We show that, if the Melnikov function has a zero of odd order and undersome further condition on the sign of the perturbation parameter, then thereexists at least one resonant solution which continues an unperturbed solution.If the Melnikov function is identically zero then one can push perturbationtheory up to the order where a counterpart of Melnikov function appears anddoes not vanish identically: if such a function has a zero of odd order and asuitable positiveness condition is met, again the same persistence result isobtained. If the system is Hamiltonian, then the procedure can be indefinitelyiterated and no positiveness condition must be required: as a byproduct, theresult follows that at least one resonant quasi-periodic solution always existswith no assumption on the perturbation. Such a solution can be interpreted as a(parabolic) lower-dimensional torus.

Resonant motions in the presence of degeneracies for quasi-periodically perturbed systems