Numerical Construction of Quasi-Periodic Solutions Beyond Symplectic Integrators

TL;DR

This work addresses the long‑time numerical computation of quasi‑periodic solutions in nearly integrable Hamiltonian systems by transforming the analytic Craig–Wayne–Bourgain framework into a practical algorithm. It introduces a dimension‑enlarged Nash‑Moser–type Newton scheme that updates frequencies via Q equations while solving nonresonant Fourier modes with an expanding lattice, yielding a priori error bounds that are independent of the total integration time. The method relies on a Gevrey‑regularity setting, a multi‑scale analysis, and careful control of small divisors through a hierarchical exclusion of resonant frequencies, culminating in a rigorous convergence theorem with super‑exponential rates. Numerically, the approach demonstrates high‑precision reconstruction of quasi‑periodic trajectories in Duffing and Henon‑Heiles systems, highlighting improved phase accuracy over standard symplectic integrators and establishing a constructive alternative to non executable global KAM arguments with potential PDE extensions.

Abstract

Symplectic integrators are the established standard for long-term simulations of nearly-integrable Hamiltonian systems due to their preservation of geometric structures. However, they suffer from an inherent limitation: secular phase-shift errors. While the qualitative ''shape'' of invariant tori is preserved, the numerical solution gradually drifts along the torus, leading to a phase-lag accumulation that degrades long-term positional accuracy. Inspired by the Craig-Wayne-Bourgain (CWB) scheme, originally developed as an analytical tool for infinite-dimensional systems, we introduce a numerical operator that incorporates frequency updates into a dimension-enlarged Newton iteration to compute quasi-periodic solutions. Unlike conventional time-stepping integrators, our alternating numerical procedure eliminates phase-lag accumulation by directly solving for instantaneous positions and phase angles. Theoretically, provided sufficient computational resources, the phase error can be reduced arbitrarily, remaining independent of the total integration time. Our algorithm translates the Nash-Moser iteration into a practical numerical framework, marking a significant departure from traditional Kolmogorov-Arnold-Moser (KAM) theory. While KAM provides rigorous existence proofs, its requirement for global Diophantine conditions and the total exclusion of resonant sets render it numerically inaccessible. By employing a ''step-by-step'' exclusion process and incrementally enlarging the dimension, our algorithm resolves irrationality conditions locally. This approach demonstrates that the ''numerical irrationality problem'' is not an intrinsic barrier to computation, offering a constructive, executable alternative to the non-executable nature of global KAM-based methods.

Figures (12)

• Figure 1: Phase space trajectories for both the exact solution of the Hamiltonian system \ref{['eqn: ham-os']} and the first-order symplectic integrator \ref{['eqn: ham-os-sym']}, starting from the initial $(x(0),y(0)) = (1,0)$; markers indicate the states at $t_n = nh$ with $n=101$.
• Figure 2: Accumulated phase (angle) error $n\Delta\theta_h$ at time $t_n = nh$ between the exact solution of the Hamiltonian system \ref{['eqn: ham-os']} and the first-order symplectic integrator \ref{['eqn: ham-os-sym']}.
• Figure 3: Schematic diagram of the numerical procedure.
• Figure 4: A diagram illustrating the alternating numerical procedure.
• Figure 5: The original small box $\Lambda_N$ and its translated counterpart $\pmb{k}_0 + \Lambda_N$.

Theorems & Definitions (31)

• Definition 2.1: Quasi-Periodic Function
• Theorem 2.2: Convergence and Error Estimation
• Remark 2.3
• Proposition 3.1
• Proposition 3.2
• Proposition 3.3
• Proposition 3.4
• Proposition 3.5
• Definition 4.1
• Lemma 4.2