Application of the consine generalized Padé approximation method in solving periodic solutions of strongly nonlinear oscillators
LI Zhenbo1,TANG Jiashi2
1. School of Mathematics and Physics, University of South China, Hengyang 421001, China;
2. College of Mechanical and Vehicle Engineering, Hunan University, Changsha 410082, China
Based on the generalized Padé approximate method, a cosine type generalized Padé approximation was constructed. According to the traits of different oscillators, the method was further modified, via which the periodic solutions of a kind of strongly nonlinear autonomous oscillators with its potential function expressed as a high order polynomial function, rational function or irrational function were obtained. Compared with numerical solutions, the precision and reliability of the proposed method were proved. In addition, the precision of the solutions keeps high in despite of that the nonlinear parameters or initial amplitude are large or small. Besides,the proposed method can be utilized in many kinds of systems, which means that the proposed method is generally applicable in wide ranges. The results show that the Padé approximate method can be utilized to solve periodic solutions directly by constructing appropriate generalized Padé appromate terms and can also provide some new considerations and reference methods.
李震波1，唐驾时2. 余弦广义Padé逼近法及其在强非线性振子周期解求解中的应用[J]. 振动与冲击, 2019, 38(22): 162-170.
LI Zhenbo1,TANG Jiashi2. Application of the consine generalized Padé approximation method in solving periodic solutions of strongly nonlinear oscillators. JOURNAL OF VIBRATION AND SHOCK, 2019, 38(22): 162-170.
[1] Chen S. H., Yang X. M., Cheung Y. K. Periodic solutions of strongly quadraticnon-linear oscillators by the elliptic perturbation method[J]. Journal of Sound and Vibration. 1998, 212: 771-780.
[2] Chen S. H., Cheung Y. K. An elliptic Lindsted-Poincare method for certain strongly non-linear oscillators[J]. Non- linear Dynamics. 1997, 12: 199-213.
[3] 陈洋洋, 燕乐纬, 陈树辉. 五次非线性自激系统的同宿解及其分岔[J]. 振动与冲击 2013, 32(11): 117-125.
Chen Y. Y, Yan L. W., Chen S. H. Homoclinic solution and bifurcation of a self-excited system with quintic strong non- linearity[J]. Journal of vibration and shock, 2013, 32(11): 117-125.
[4] Li Z. B., Tang J. S., Cai P. A generalized harmonic function perturbation method for determining limit cycles and homoclinic orbits of Helmholtz—Duffing oscillator[J]. Journal of Sound and Vibration, 2013, 332: 5508-5522.
[5] Li Z. B., Tang J. S., Cai P. Predicting homoclinic and heteroclinic bifurcation of generalized Duffing-Harmonic-van de Pol oscillator[J]. Qualitative Theory of Dynamical Systems, 2016, 15:19-37；
[6] Chen Y Y, Chen S H, Zhao W. Constructing explicit homoclinic solution of oscillators: An improvement for perturbation procedure based on nonlinear time transformation[J]. Communications in Nonlinear Science and Numerical Simulation, 2017, 48:123–139
[7] Chen Y Y, Yan L W, Ray Kai-Leung Su, Liu B. Generalization of hyperbolic perturbation solution for heteroclinic orbits of strongly nonlinear self-excited oscillator[J]. Journal of Vibration and Control, 2017, 23(19): 3071-3091
[8] Liao S. J. Homotopy analysis method: A new analytical technique for nonlinear problems[J]. Communications in Nonlinear Science and Numerical Simulation, 1997, 2: 95-100.
[9] 郑敏毅, 胡辉, 郭源君, 孙光永. 应用优化的同伦分析法求解非线性Jerk方程[J]. 振动与冲击, 2012, 31(5): 21-25.
Zheng M. Y., Hu H., Guo Y. J., Sun G. Y. Optimal homotopy analysis method applied to solve a nonlinear Jerk eqaution[J]. Journal of vibration and shock, 2012, 31(5): 21-25.
[10] Liu Q X, Liu J K, Chen Y M. Asymptotic limit cycle of fractional van der Pol oscillator by homotopy analysis method and memory-free principle[J]. Applied Mathematical Modelling, 2016, 40(4): 3211-3220
[11] He J. H. A new approach to nonlinear partial differential equations[J]. Communications in Nonlinear Science and Numerical Simulation, 1997, 2: 230-235.
[12] Cao Y. Y., Chung K. W., Xu J. A novel construction of homoclinic and heteroclinic orbits in nonlinear oscillators by a perturbation-incremental method[J]. Nonlinear Dynamics. 2011, 64: 221-236.
[13] Emaci E, Vakakis A F, Andrianov I V, Mikhlin Y U. Study of two-dimensional axisymmetric breathers using Padé approximants [J]. Nonlinear Dynamics, 1997, 13:327-338
[14] Mikhlin Y. V. Analytical construction of homoclinic orbits of two-and three-dimensional dynamic systems[J]. Journal of Sound and Vibration, 2000, 230: 971-983.
[15] 张琪昌, 冯晶晶, 王炜. 类Padé逼近方法在二维振动系统的应用[J]. 力学学报, 2011, 43(5): 914-921.
Zhang Q. C, Feng J. J, Wang W. The construction of homoclinic and heterclinic orbit in two-dimensional nonlinear system based on the quasi-Padé approximation[J]. Chinese Journal of Theoretical and Applied Mechanics, 2011, 43(5): 914-921.
[16] Feng J. J, Zhang Q. C, Wang W. Chaos of several typical asymmetric systems[J]. Chaos, Solitons & Fractals. 2012, 45: 950-958.
[17] 李震波, 唐驾时, 蔡萍. 求强非线性自治振子同宿轨的广义Padé逼近方法[J]. 力学学报, 2013, 45(3): 461-464.
Li Z. B., Tang J. S., Cai P. Homoclinic orbit of strongly nonlinear autonomous oscillator via generalized padé approximation method [J]. Chinese Journal of Theoretical and Applied Mechanics, 2013, 45(3): 461-464.
[18] 徐献瑜, 李家楷, 徐国良.Padé逼近概论[M]. 上海: 上海科学技术出版社, 1990.
Xu X. Y., Li J. K., Xu G. L. Introduction to Padé Approximation[J]. Shanghai: Shanghai Science and Technology Press, 1990.
[19] Baker G.A. Essential of Padé Approximants[M]. New York: Academic, 1975.
[20] Baker G.A., Gammel J. L. The Padé Approximant in theo- retical physics[M]. New York: Academic Press, 1970.
