全部 标题 作者
关键词 摘要

OALib Journal期刊
ISSN: 2333-9721
费用：99美元
投递稿件

查看量下载量

相关文章

更多...

一类含周期激励Duffing系统的数值模拟
Numerical Simulation of a Class of Duffing Systems with Periodic Excitation

DOI: 10.12677/DSC.2021.101009, PP. 77-90

Keywords: Duffing系统，阻尼系数，混沌现象，四阶Runge-Kutta算法
Duffing System, Damping Coefficient, Chaotic Phenomenon, Fourth-Order Runge-Kutta Algorithm

Full-Text   Cite this paper   Add to My Lib

Abstract:

本文主要探究一类有周期激励的Duffing系统，根据Melnikov方法、欧拉离散、四阶Runge-Kutta算法理论及Matlab程序等进行数值模拟。从理论上分析不同参数下系统的动力学现象，如分岔、周期运动、混沌运动等，作出时间序列图、相图、分岔图及最大Lyapunov指数图，观测在振幅、频率及阻尼项变化时Duffing系统产生的复杂动态行为。数值模拟发现，在考虑激励振幅和频率不变时，随阻尼项的增大，系统混沌行为会发生显著变化，混沌状态将逐渐消失形成周期运动，而在考虑振幅影响时，系统混沌状态则无明显改变。
This paper mainly explores a Duffing system with periodic excitation. Based on Melnikov method, Euler discretization, fourth-order Runge-Kutta algorithm theory and Matlab program, it theoretically analyzes the dynamic phenomena of the system under different parameters, such as bifurcation, periodic motion and chaos, makes time series diagram, phase diagram, bifurcation diagram and maximum Lyapunov index diagram, and observes Duffing system when amplitude, frequency and damping terms change. Numerical simulation shows that when the excitation amplitude and frequency are constant, with the increase of damping term, the chaotic behavior of the system will change significantly, and the chaotic state will gradually disappear to form periodic motion, while when the influence of amplitude is considered, the chaotic state of the system will not change much.

References

[1]  Li, M.D., Yuan, Z.C. and Luo, D.B. (2020) In-Depth Study of Chaos in a Nonlinear Dynamic System. University Physics, 39, 33-37.
[2]  Li, Q.D. and Yang, X.S. (2012) Progresses on Chaotic Dynamics Study with Topological Horseshoes. Journal of Dynamics & Control, 4, 293-298.
[3]  Cai, M.X., Yang, J.P. and Deng, J. (2014) Bifurcations and Chaos in Duffing Equation with Damping and External Excitations. Acta Mathematicae Applicatae Sinica, English Series, 30, 483-504.
https://doi.org/10.1007/s10255-014-0284-0
[4]  Fernando, A., Ana, P.S.D. and Carla, M.A.P. (2010) Quasi-Periodic States in Coupled Rings of Cells. Communications in Nonlinear Science and Numerical Simulation, 15, 1048-1062.
https://doi.org/10.1016/j.cnsns.2009.05.035
[5]  Tao, J. and Zhi, Y.Y. (2017) Bifurcations and Chaos in the Duffing Equation with Parametric Excitation and Single External Forcing. International Journal of Bifurcation & Chaos, 27, Article ID: 1750125.
https://doi.org/10.1142/S0218127417501255
[6]  Plaut, R.H. and Hsich, J.C. (1987) Chaos in a Mechanism with Time Delays under Parametric and External Excitation. Journal of Sound and Vibration, 114, 73-90.
https://doi.org/10.1016/S0022-460X(87)80235-3
[7]  Raghothama, A. and Narayanan, S. (2002) Periodic Response and Chaos in Nonlinear Systems with Parametric Excitation and Time Delay. Nonlinear Dynamics, 27, 341-365.
https://doi.org/10.1023/A:1015207726565
[8]  刘延柱, 陈立群. 非线性振动[M]. 北京: 高等教育出版社, 2011: 224-228.
[9]  Zuo, Z.L. and Yu, X. (2019) A Design Method of Chaotic Synchronous Multi-Stable Manifold. Materials Science and Engineering, 544, 12-37.
https://doi.org/10.1088/1757-899X/544/1/012037
[10]  He, H.J., Cui, Y. and Sun, G. (2019) Dynamic Analysis and Chaos Control of a New Nonlinear System. Journal of Jilin University, 57, 1224-1230.
[11]  李庆, 关治, 白峰山. 数值计算原理[M]. 北京: 清华大学出版社, 2004.
[12]  Huang, X.-R. and Jézéquel, L.B. (2018) Nonlinear Modal Synthesis for Analyzing Structures with a Frictional Interface Using a Generalized Masing Model. Journal of Sound & Vibration, 434, 166-191.
https://doi.org/10.1016/j.jsv.2018.07.027
[13]  Qian, Y.H. and Yan, D.M. (2018) Fast-Slow Dynamics Analysis of a Coupled Duffing System with Periodic Excitation. International Journal of Bifurcation & Chaos in Applied Sciences & Engineering, 28, Article ID: 1850148.
https://doi.org/10.1142/S0218127418501481
[14]  Zheng, J.K., Zhang, X.F. and Bi, Q.S. (2019) Clusters Oscillation and Delayed Fork Bifurcation in a Class of Chaotic Systems. Chinese Journal of Theoretical and Applied Mechanics, 51, 540-549.
[15]  Cai, M.X. and Yang, J.P. (2006) Bifurcation of Periodic Orbits and Chaos in Duffing Equation. Acta Mathematicae Applicatae Sinica, English Series, 22, 495-508.
https://doi.org/10.1007/s10255-006-0325-4
[16]  Ji, J.C. and Leung, A.Y. (2002) Bifurcation Control of a Parametrically Excited Duffing System. Nonlinear Dynamics, 27, 411-417.
https://doi.org/10.1023/A:1015221422293
[17]  Gong, S. and Wang, X.Yu. (2019) Dynamic Analysis of Vibrating Flip-Flow Screen Based on a Nonlinear Model of Shear Spring. Journal of the China Coal Society, 44, 3241-3249.
[18]  Chen, Y.N., Meng, W.J. and Qian, Y.H. (2020) Fixed Point Chaos and Fold/Fold Bursting of a Class of Duffing Systems and the Mechanism Analysis. Chinese Journal of Theoretical and Applied Mechanics, 52, 1475-1484.
[19]  Li, X.H., et al. (2019) New Periodic-Chaotic Attractors in Slow-Fast Duffing System with Periodic Parametric Excitation. Scientific Reports, 9, Article No. 11185.
https://doi.org/10.1038/s41598-019-46768-7

Full-Text

Contact Us

service@oalib.com

QQ:3279437679

WhatsApp +8615387084133