Solution procedure of residue harmonic balance method and its applications

• Published in: Science China. Physics, mechanics & astronomy mechanics & astronomy, 2014-08, Vol.57 (8), p.1581-1591
• Main Authors: Guo, ZhongJin, Leung, A. Y. T., Ma, XiaoYan
• Format: Article
• Language: English
• Subjects: Amplitudes; Approximation; Astronomy; Classical and Continuum Physics; Differential equations; Exact solutions; Harmonic balance method; Observations and Techniques; Ordinary differential equations; Parameters; Physics; Physics and Astronomy; Residues; System effectiveness; 周期频率; 常微分方程组; 应用程序; 微分自治系统; 残留; 线性常微分方程; 谐波平衡法; 高次谐波
• ISSN: 1674-7348; 1869-1927
• DOI: 10.1007/s11433-013-5317-9

This paper presents a simple and rigorous solution procedure of residue harmonic balance for predicting the accurate approxi- mation of certain autonomous ordinary differential systems. In this solution procedure, no small parameter is assumed. The harmonic residue of balance equation is separated in two parts at each step. The first part has the same number of Fourier terms as the present order of approximation and the remaining part is used in the subsequent improvement. The corrections are governed by linear ordinary differential equation so that they can be solved easily by means of harmonic balance method again. Three kinds of different differential equations involving general, fractional and delay ordinary differential systems are given as numerical examples respectively. Highly accurate limited cycle frequency and amplitude are captured. The results match well with the exact solutions or numerical solutions for a wide range of control parameters. Comparison with those available shows that the residue harmonic balance solution procedure is very effective for these autonomous differential systems. Moreover, the present method works not only in predicting the amplitude but also the frequency of bifurcated period solution for delay ordinary differential equation.