Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2604.04955

Effective stability estimates close to resonances with applications to rotational dynamics

Alessandra Celletti·Anargyros Dogkas·Alessia Francesca Guido
View original source ↗Read source document ↗Export metadata ↓

Abstract

We consider nearly-integrable Hamiltonian systems defined over a non-resonant domain. In the neighborhood of resonances, we use Nekhoroshev-like estimates to provide effective stability bounds for the action variables over long time. The applicability conditions of these estimates allow some freedom in the choice of parameters. Hence, we develop an optimization algorithm for choosing parameters that maximize the stability time. To further improve the stability estimates, we use perturbation theory to reduce the norm of the perturbing function. We implement this procedure (effective stability estimates and perturbation theory) to analyze the stability of sequences of irrational (Diophantine) frequencies converging to frequencies corresponding to resonances. We consider two applications to models describing problems of rotational dynamics in Celestial Mechanics: the spin-orbit problem, described by a 1D time-dependent Hamiltonian, and the spin-spin-orbit model, described by a 2D time-dependent Hamiltonian. We show stability results for orbits close to the main resonances associated with such models.

Explore related subjects

math.DSastro-ph.EPmath-phmath.MP

Keep this discovery

BibTeXRIS

Alessandra Celletti, Anargyros Dogkas, Alessia Francesca Guido. 2026-04-03. Effective stability estimates close to resonances with applications to rotational dynamics. https://arxiv.org/abs/2604.04955

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Admissible Fourier Lengths, KAM Reducibility, and Spectral Applications

We develop a perturbative KAM reducibility theory for one-frequency $\mathrm{SL}(2,\mathbb{R})$ cocycles based on an admissible Fourier length $\ell$. The regularity relevant to the iteration is measured by positive adapted Fourier width rather than ordinary smoothness in the Euclidean length $|n|$. The same length governs Fourier decay, truncation and resonance scales, and the arithmetic condition controlling the small divisors. This framework contains the classical analytic and Gevrey settings, while non-monotone choices of $\ell$ allow classical nowhere differentiable Weierstrass-type perturbations and continuous perturbations outside every positive H\"older class. As spectral applications, we obtain purely absolutely continuous spectrum for every phase and $1/2$-H\"older continuity of the integrated density of states for the associated quasiperiodic Schr\"odinger operators. The Aubry dual has pure point spectrum for Lebesgue almost every dual phase, with eigenfunctions exponentially localized in the metric induced by $\ell$. We also construct nowhere differentiable quasiperiodic potentials with purely absolutely continuous Cantor spectrum.

math.DS↗
arXiv2026

Dynamics inside the attracting basins of some skew products

Polynomial skew products in $\mathbb{C}^2$ are maps of the form $F(z,w)=(P(z),Q(z,w))$, where $P$ and $Q$ are polynomials. Their local dynamics have been widely investigated. In this paper, we study the global dynamics inside Fatou components of some skew products. We consider all the inverse images in a Fatou component of a given point and use the Kobayashi metric to measure the distance between points. In the cases we consider, there are always arbitrarily large Kobayashi balls in the complement of these inverse sets.

math.DS↗
arXiv2026

Ergodicity of dynamical systems without uniqueness of orbits

Recently, there has been considerable interest in the study of non-deterministic dynamical systems. To analyze the chaotic behavior of such systems from a measure-theoretic viewpoint, it is desirable to consider ergodicity. However, the classical definition of ergodicity involves invariant sets, whose definition is not unique for non-deterministic dynamical systems. Thus, we are led to the question of which invariance yields an interesting definition of ergodicity. Here, we propose a definition based on the strong backward invariance and show that analogs of classical results hold. We also consider implications of the Birkhoff ergodic theorem for systems without uniqueness of orbits.

math.DS↗