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

TL;DR

This paper introduces a unified multi-steps variable-coefficient formulation (m-VCF) for computing and tracking quasi-periodic solutions with multiple base frequencies using harmonic balance, collocation, or finite difference methods, along with an efficient nonlinear force evaluation and a new phase condition.

Contribution

It proposes a novel unified framework (m-VCF) and an alternating U and S domain method (AUS) for solving and tracking multi-frequency quasi-periodic solutions, addressing previous challenges.

Findings

Validated on three nonlinear systems demonstrating effectiveness.

Successfully tracks solutions with unknown multiple base frequencies.

Assesses stability using Lyapunov exponents.

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.