Torus Time-Spectral Method for Quasi-Periodic Problems
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908713388867584 |
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| author | He, Sicheng Li, Hang Ekici, Kivanc |
| author_facet | He, Sicheng Li, Hang Ekici, Kivanc |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_13631 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Torus Time-Spectral Method for Quasi-Periodic Problems He, Sicheng Li, Hang Ekici, Kivanc Numerical Analysis Dynamical Systems 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. |
| title | Torus Time-Spectral Method for Quasi-Periodic Problems |
| topic | Numerical Analysis Dynamical Systems |
| url | https://arxiv.org/abs/2512.13631 |