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arXiv · 2512.13631

Torus Time-Spectral Method for Quasi-Periodic Problems

Sicheng He·Hang Li·Kivanc Ekici
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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.

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Sicheng He, Hang Li, Kivanc Ekici. 2025-12-15. Torus Time-Spectral Method for Quasi-Periodic Problems. https://arxiv.org/abs/2512.13631

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