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arXiv · 2505.02021

General multi-steps variable-coefficient formulation for computing quasi-periodic solutions with multiple base frequencies

Junqing Wu·Ling Hong·Mingwu Li·Jun Jiang
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Abstract

Quasi-periodic solutions with multiple base frequencies exhibit the feature of $2\pi$-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.

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Junqing Wu, Ling Hong, Mingwu Li, Jun Jiang. 2025-05-04. General multi-steps variable-coefficient formulation for computing quasi-periodic solutions with multiple base frequencies. https://doi.org/10.1007/s10409-025-25381-x

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