Numerical Construction of Elliptic Lower-Dimensional Quasi-Periodic Solutions with a Priori Bound

TL;DR

This paper develops a numerical scheme to construct lower-dimensional quasi-periodic solutions in nearly integrable systems, overcoming resonance and small divisor issues, with applications demonstrated on classical models.

Contribution

It extends an existing alternating numerical scheme to effectively compute elliptic lower-dimensional quasi-periodic solutions considering complex resonance conditions.

Findings

Successfully applied to Hénon-Heiles and FPU models.

Demonstrates the effectiveness of the extended scheme in challenging resonance scenarios.

Shows the perturbation operator has Gevrey decay without Hankel structure.

Abstract

A numerical framework for constructing full-dimensional quasi-periodic solutions in nearly integrable systems was recently developed by Fu and Shi[2026]. Based on an alternating scheme, this approach effectively overcomes the secular drift in angle variables, a fundamental limitation of symplectic integrators. However, in many applications, such as the restricted three-body problem, lower-dimensional quasi-periodic solutions hold greater significance. The construction of these solutions is considerably more challenging due to the presence of normal frequencies, leading to intricate resonance phenomena. Beyond the subspace resonance, one must also account for the first and second Melnikov conditions to eliminate small divisors. In this study, we extend the proposed alternating numerical scheme to compute the elliptic lower-dimensional quasi-periodic solutions. Numerical experiments are presented for the H\'{e}non-Heiles model and the Fermi--Pasta--Ulam (FPU) model, demonstrating the effectiveness of the proposed method. Furthermore, we emphasize that the perturbation is not merely a polynomial with real coefficients but is a real-valued function. As a result, the associated perturbation operator exhibits Gevrey decay without possessing a Hankel structure. Meanwhile, we further simplify the multi-scale analysis by exploiting the resolvent identity, showing that the global inverse can be expressed linearly in terms of local inverses via the gluing procedure. This representation reveals a regime-dependent interaction structure: weak interactions dominate at short range, while strong interactions emerge at long range. This balance ensures that the Gevrey decay of the inverse remains uniformly controlled. Moreover, within this linear representation, the inversion conditions provide a clearer characterization of the localization properties.