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

TL;DR

The paper addresses the challenge of computing quasi-periodic solutions with multiple base frequencies by introducing a unified multi-steps variable-coefficient formulation (m-VCF) that can utilize harmonic balance, collocation, or finite difference methods within a hyper-time domain. Central to the approach is the Alternating U and S domain (AUS) for efficient evaluation of nonlinear term contributions, together with a robust phase condition to fix unknown base frequencies and a continuation scheme to track solutions. The framework yields a set of nonlinear algebraic equations whose zeros correspond to the desired quasi-periodic solutions, with stability analyzed via Lyapunov exponents and NS-induced torus formation. Validation on three nonlinear systems demonstrates consistent multi-frequency responses, effective initialization after bifurcations, and reliable stability assessment, highlighting the practical potential of m-VCF for complex dynamical systems.

Abstract

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.

Figures (15)

• Figure 1: FRCs for the periodic (d=1) and quasi-periodic (d=2,3) orbits of the Duffing-van der Pol oscillator.
• Figure 2: The quasi-periodic solution in the S domain for $\omega_1=2.65$, Phase diagrams: a) d=1, c) d=2, e) d=3; Poincare sections: b) d=1, d) d=2, f) d=3.
• Figure 3: The comparisons of the quasi-periodic solutions computed by the presented method and TI in the frequency domain: a) d=1, b) d=2, c) d=3.
• Figure 4: a): FRCs for the quasi-periodic (d=2) orbits of the Duffing-van der Pol oscillator, b), c): Zoom of peaks.
• Figure 5: A cantilevered pipe conveying fluid subjected to a periodic excitation.