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.

Lower-Dimensional Invariant Tori for Perturbations of a Class of Non-convex Hamiltonian Functions

“…Our aim is to represent the formal series as a "sum over trees", so first of all we need some definitions. (We closely follow [3,4], with obvious adaptations).…”

“…Set χ −1 (x) = 1 and χ n (x) = χ(8x/α mn (ω)) for n ≥ 0. Set also ψ(x) = 1 − χ(x), ψ n (x) = ψ(8x/α mn (ω)), and Ψ n (x) = χ n−1 (x)ψ n (x), for n ≥ 0; see Figure 3.5 in [4]. Then we associate with each line a propagator…”

On the continuation of degenerate periodic orbits via normal form: full dimensional resonant tori

“…In the same spirit, variational methods which make use of the mountain pass theorem were developed some years later, by Fadell and Rabinowitz, under different hypothesis (see Chapter 1 in [4] for a simplified but extremely clear introduction to this result). In the last years, some new results appeared [5,6,18,28], which face the problem of the continuation of degenerate periodic orbits and lower dimensional tori in weakly perturbed Hamiltonian systems.…”

Resonant tori of arbitrary codimension for quasi-periodically forced systems

“…Such assumption can be seen as a time-reversibility condition on the Hamiltonian system. For non-convex Hamiltonians (1.3) with r = 1, the same result of persistence of at least one invariant torus has been proved in [11] without assuming Hypothesis 2 on the perturbation; the result has been extended to more general one-dimensional systems in [12]. Strictly speaking, Cheng's result does not imply the result in [11,12], because the unperturbed Hamiltonian is not convex; however Cheng's method can be adapted to such a case; see [13] for an explicit implementation.…”

“…Proof. The first identity can be proved as Lemma 4.8 in [12]. The second one is easier: integrating by parts and using Lemma 3.7 and the fact that [b ≤p (·; ε, β 0 )] 0 = 0, we obtain, shortening again b ≤p = b ≤p (α; ε, β 0 ),…”