Moser's theorem for lower dimensional tori

Abstract

Moser’s C-version of Kolmogorov’s theorem on the persistence of maximal quasi-periodic\ud solutions for nearly-integrable Hamiltonian system is extended to the persistence of non-maximal\ud quasi-periodic solutions corresponding to lower-dimensional elliptic tori of any dimension n\ud between one and the number of degrees of freedom. The theorem is proved for Hamiltonian\ud functions of class C for any >6n + 5 and the quasi-periodic solutions are proved to be of\ud class Cp for any p with 2<p<p∗ for a suitable p∗ = p∗(n, )>2 (which tends to infinity\ud when →∞)

A renormalization approach to lower-dimensional tori with Brjuno frequency vectors

“…The proof of existence of elliptic (stable) lower-dimensional tori in the general case was given by Eliasson [6]. For an overview of the results on lower-dimensional tori, the reader is referred to [4,[6][7][8][9][10]22,24,25,27,29,32]. Lowerdimensional tori also appear in and are particularly relevant for PDEs (see e.g.…”

On lower dimensional invariant tori in C d reversible systems

“…Such approach applies to some PDEs with periodic boundary condition (see [2]). Especially, when the small perturbations f i (i = 1, 2, 3, 4) are functions of class C d (d > 6n + 5) and normal frequencies Ω i are simple, Chierchia and Qian [3] showed the persistence and regularity of the lower n-dimensional elliptic tori.…”

On the existence of invariant tori in non-conservative dynamical systems with degeneracy and finite differentiability

“…In the context of finitely differentiable perturbations, the study on the persistence of quasi-periodic invariant tori has originated from the work of Moser [24] on area-preserving mappings of an annulus, which was extended to dissipative vector fields in [6] based on smoothing operator technique. Another important method, which can relax the requirement for regularity of perturbations, is to approximate a differentiable function by real analytic ones [25,30,40,27,35,13,1,38]. Rüssmann [30] proved an optimal estimate result on approximating a differentiable function by analytic ones.…”

“…Following this approach Zehnder [40] established a generalized implicit function theorem and applied it to the existence of parameterized invariant tori of nearly integrable Hamiltonian systems in finitely differentiable case, Pöschel [27] showed that on a Cantor set, invariant tori of the perturbed Hamiltonian system form a differentiable family in the sense of Whitney. The results and ideas of Moser and Pöschel are extended to the case of symplectic mappings by Shang [35] and to the case of lower dimensional elliptic tori by Chierchia and Qian [13], respectively. Wagener [38] extended the modifying terms theorem of Moser [26] (i.e., introducing additional parameters) to finitely differentiable and Gevrey regular vector fields.…”

“…The results mentioned above, except for [13], were restricted to the case where the integrable part is analytic in coordinate variables as well as in parameters. The integrable part in [13] is assumed to be Lipschitz with respect to parameters and the frequency map to be a Lipschitz homeomorphism.…”