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

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 is proposed, which provides a unified framework to enable either harmonic balance method or collocation method or finite difference method to solve quasi-periodic solutions with multiple base frequencies. For this purpose, a method of alternating U and S domain 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.