Torus Time-Spectral Method for Quasi-Periodic Problems

TL;DR

This paper introduces a torus time-spectral method that efficiently solves quasi-periodic problems by extending the phase space and applying double-Fourier collocation, demonstrating spectral convergence and broad applicability.

Contribution

The paper presents a novel torus time-spectral method that generalizes Fourier-based approaches to handle quasi-periodic systems with rigorous convergence proof.

Findings

Spectral error decay observed on the torus

Method shows tight agreement with time-accurate integrations

Efficient for higher-dimensional tori and multi-frequency dynamics

Abstract

Quasi-periodic trajectories with two or more incommensurate frequencies are ubiquitous in nonlinear dynamics, yet the classical Fourier-based time-spectral method is tied to strictly periodic responses. We introduce a torus time-spectral method that lifts the governing equations to an extended angular phase space, applies double-Fourier collocation on the invariant torus, and solves for the state. The formulation exhibits spectral convergence for quasi-periodic problem which we give a rigorous mathematical proof and also verify numerically. We demonstrate the approach on Duffing oscillators and a nonlinear Klein-Gordon system, documenting spectral error decay on the torus and tight agreement with time-accurate integrations while using modest frequency grids. The method extends naturally to higher-dimensional tori and offers a computationally efficient framework for analyzing quasi-periodic phenomena in fluid mechanics, plasma physics, celestial mechanics, and other domains where multi-frequency dynamics arise.