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

TL;DR

This paper investigates the existence of resonant quasi-periodic solutions in one-dimensional systems with quasi-periodic perturbations, extending Melnikov theory to degenerate cases and Hamiltonian systems, with implications for lower-dimensional tori.

Contribution

It generalizes Melnikov theory to systems with degeneracies and Hamiltonian structures, establishing conditions for the persistence of resonant solutions without sign restrictions.

Findings

Resonant solutions exist under odd-order zeroes of the Melnikov function.

The method applies iteratively to Hamiltonian systems without sign conditions.

At least one resonant quasi-periodic solution always exists in Hamiltonian cases.

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.