**Author:**
Livia Corsi and Guido Gentile

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

**Abstract: **
We consider one-dimensional systems in the presence of a quasi-periodic perturbation, in the
analytical setting, and study the problem of existence of quasi-periodic solutions which are resonant
with the frequency vector of the perturbation. We assume that the unperturbed system
is locally integrable and anisochronous, and that the frequency vector of the perturbation satisfies
the Bryuno condition. Existence of resonant solutions is related to the zeroes of a suitable
function, called the Melnikov function, by analogy with the periodic case. We show that, if
the Melnikov function has a zero of odd order and under some further condition on the sign of
the perturbation parameter, then there exists at least one resonant solution which continues an
unperturbed solution. If the Melnikov function is identically zero then one can push perturbation
theory up to the order where a counterpart of Melnikov function appears and does not vanish
identically: if such a function has a zero of odd order and a suitable positiveness condition is met,
again the same persistence result is obtained. If the system is Hamiltonian, then the procedure
can be indefinitely iterated and no positiveness condition must be required: as a byproduct, the
result follows that at least one resonant quasi-periodic solution always exists with no assumption
on the perturbation. Such a solution can be interpreted as a (parabolic) lower-dimensional torus.

**Keywords:**
Quasiperiodic motions;
Quasi-periodically forced systems;
Renormalization group;
Bryuno vectors;
Bifurcations;
Small divisors;
Melnikov problem;
Lower-dimensional tori.

Livia Corsi

Dipartimento di Matematica

Università di Roma Tre

Largo San Leonardo Murialdo 1, 00146 Roma, Italy

e-mail: lcorsi@mat.uniroma3.it

Guido Gentile

Dipartimento di Matematica

Università di Roma Tre

Largo San Leonardo Murialdo 1, 00146 Roma, Italy

e-mail: gentile@mat.uniroma3.it