General multi-steps variable-coefficient formulation for computing quasi-periodic solutions with multiple base frequencies
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866913819609006080 |
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| author | Wu, Junqing Hong, Ling Li, Mingwu Jiang, Jun |
| author_facet | Wu, Junqing Hong, Ling Li, Mingwu Jiang, Jun |
| contents | Quasi-periodic solutions with multiple base frequencies exhibit the feature of $2π$-periodicity with respect to each of the hyper-time variables. However, it remains a challenge work, due to the lack of effective solution methods, to solve and track the quasi-periodic solutions with multiple base frequencies until now. In this work, a multi-steps variable-coefficient formulation (m-VCF) is proposed, which provides a unified framework to enable either harmonic balance method (HB) or collocation method (CO) or finite difference method (FD) to solve quasi-periodic solutions with multiple base frequencies. For this purpose, a method of alternating U and S domain (AUS) is also developed to efficiently evaluate the nonlinear force terms. Furthermore, a new robust phase condition is presented for all of the three methods to make them track the quasi-periodic solutions with prior unknown multiple base frequencies, while the stability of the quasi-periodic solutions is assessed by mean of Lyapunov exponents. The feasibility of the constructed methods under the above framework is verified by application to three nonlinear systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_02021 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | General multi-steps variable-coefficient formulation for computing quasi-periodic solutions with multiple base frequencies Wu, Junqing Hong, Ling Li, Mingwu Jiang, Jun Dynamical Systems Quasi-periodic solutions with multiple base frequencies exhibit the feature of $2π$-periodicity with respect to each of the hyper-time variables. However, it remains a challenge work, due to the lack of effective solution methods, to solve and track the quasi-periodic solutions with multiple base frequencies until now. In this work, a multi-steps variable-coefficient formulation (m-VCF) is proposed, which provides a unified framework to enable either harmonic balance method (HB) or collocation method (CO) or finite difference method (FD) to solve quasi-periodic solutions with multiple base frequencies. For this purpose, a method of alternating U and S domain (AUS) is also developed to efficiently evaluate the nonlinear force terms. Furthermore, a new robust phase condition is presented for all of the three methods to make them track the quasi-periodic solutions with prior unknown multiple base frequencies, while the stability of the quasi-periodic solutions is assessed by mean of Lyapunov exponents. The feasibility of the constructed methods under the above framework is verified by application to three nonlinear systems. |
| title | General multi-steps variable-coefficient formulation for computing quasi-periodic solutions with multiple base frequencies |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2505.02021 |