Approximate analytical solution in slow-fast system based on modified multi-scale method

• Published in: Applied mathematics and mechanics Vol. 41; no. 4; pp. 605 - 622
• Main Authors: Li, Xianghong, Tang, Jianhua, Wang, Yanli, Shen, Yongjun
• Format: Journal Article
• Language: English
• Published: Shanghai Shanghai University 01.04.2020; Springer Nature B.V; Department of Mathematics and Physics, Shijiazhuang Tiedao University,Shijiazhuang 050043, China%Department of Mechanical Engineering, Shijiazhuang Tiedao University,Shijiazhuang 050043, China
• Edition: English ed.
• Subjects: Applications of Mathematics; Classical Mechanics; Exact solutions; Fluid- and Aerodynamics; Harmonic balance method; Harmonic excitation; Mathematical analysis; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Multiscale analysis; Nonlinear analysis; Oscillators; Partial Differential Equations; Resonant frequencies; Spiking; van der Pol system; O322; 34E13; modified multi-scale method (MMSM); 74H10; slow-fast system; multi-scale method (MSM)
• ISSN: 0253-4827; 1573-2754
• DOI: 10.1007/s10483-020-2598-9

A simple, yet accurate modified multi-scale method (MMSM) for an approximately analytical solution in nonlinear oscillators with two time scales under forced harmonic excitation is proposed. This method depends on the classical multi-scale method (MSM) and the method of variation of parameters. Assuming that the forced excitation is a constant, one could easily obtain the approximate analytical solution of the simplified system based on the traditional MSM. Then, this solution for the oscillator under forced harmonic excitation could be established after replacing the harmonic excitation by the constant excitation. To certify the correctness and precision of the proposed analytical method, the van der Pol system with two scales subject to slowly periodic excitation is investigated; this system presents rich dynamical phenomena such as spiking (SP), spiking-quiescence (SP-QS), and quiescence (QS) responses. The approximate analytical expressions of the three types of responses are given by the MMSM, and it can be found that the precision of the new analytical method is higher than that of the classical MSM and better than that of the harmonic balance method (HBM). The results obtained by the present method are considerably better than those obtained by traditional methods, quantitatively and qualitatively, particularly when the excitation frequency is far less than the natural frequency of the system.