机械动力学:2017,Vol:36,Issue(9):1340-1344
引用本文:
张艳龙, 石建飞, 王丽. 双参数平面上Duffing系统TLE的计算与分岔分析[J]. 机械科学与技术
Zhang Yanlong, Shi Jianfei, Wang Li. Calculating Top Lyapunov Exponent of Duffing System on Two-parameter Plane and Analyzing its Bifurcation[J]. Journal Of Remote Sensing

双参数平面上Duffing系统TLE的计算与分岔分析
张艳龙1, 石建飞1, 王丽2
1. 兰州交通大学机电工程学院, 兰州 730070;
2. 兰州城市学院数学学院, 兰州 730070
摘要:
给出系统在参数空间最大Lyapunov指数的计算方法,计算Duffing系统在双参数平面上最大Lyapunov指数的分布特性。结合单参数分岔图讨论了Duffing系统在双参数平面上的分岔特性。结果表明系统在双参数平面上出现了周期跳跃、叉式分岔和倍周期分岔等各种分岔曲线,系统在倍周期分岔曲线环内不断嵌套新的倍周期分岔曲线环,使得系统最终经倍周期分岔序列进入混沌运动;这些倍周期分岔曲线环均被周期跳跃曲线截断,使得系统经过周期跳跃曲线后出现不同的周期运动。参数平面上各种分岔曲线的相交使得系统局部分岔特性变得极为复杂。通过对Duffing系统的计算与分析证明了本文方法在计算混沌问题方面的有效性与可行性。
关键词:    Duffing系统    最大Lyapunov指数    双参数特性    分岔    周期跳跃   
Calculating Top Lyapunov Exponent of Duffing System on Two-parameter Plane and Analyzing its Bifurcation
Zhang Yanlong1, Shi Jianfei1, Wang Li2
1. School of Mechanical Engineering, Lanzhou Jiaotong University, Lanzhou 730070, China;
2. School of Mathematics, Lanzhou City University, Lanzhou 730070, China
Abstract:
The method of calculating the top Lyapunov exponent of a Duffing system in the multi-parameter space is given, and the distribution characteristic of the top Lyapunov exponent of the Duffing system in its two-parameter plane is calculated. Bifurcation characteristics of the Duffing system are discussed with the single parameter bifurcation diagram. The discussion results show that the Duffing system has various bifurcation curves including periodic jump, pitchfork bifurcation and period-doubling bifurcation in its two-parameter plane. The period-doubling bifurcation curve cycles constantly appear and nest each other, making the system finally evolve the chaotic state via period-doubling bifurcation sequences. These periodic bifurcation curves are truncated by the periodic jumping curve, making the system move into different periodic states via the cycle jump curve. The intersection of various bifurcation curves in the two-parameter plane makes the local bifurcation characteristic of the system become very complex. The calculation and analysis of the Duffing system prove that this method is effective and feasible in terms of computational chaos.
Key words:    Duffing system    top Lyapunov exponent    two-parameter plane    bifurcation    periodic jump   
收稿日期: 2016-09-05     修回日期:
DOI: 10.13433/j.cnki.1003-8728.2017.0905
基金项目: 国家自然科学基金项目(11302092,11362008)资助
通讯作者:     Email:
作者简介: 张艳龙(1981-),副教授,博士研究生,研究方向为动力学及控制,zhangyl@mail.lzjtu.cn
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参考文献:
[1] 洪灵,徐健学.两参量平面上双重激变尖点研究[J].物理学报,2002,51(12):2694-2701 Hong L, Xu J X. A study on the double crisis vertex in a two-parameter plane[J]. Acta Physica Sinica, 2002,51(12):2694-2701(in Chinese)
[2] 孙清,张斌,刘正伟,等.含双时滞振动主动控制系统超谐共振及亚谐共振分析[J].工程力学,2010,27(12):84-89 Sun Q, Zhang B, Liu Z W, et al. Analysis of superharmonic and subharmonic resonance responses of active vibration control system with double time delay[J]. Engineering Mechanics,2010,27(12):84-89(in Chinese)
[3] 吴志强,张振华,郝颖.双线性双滞后环系统的约束分岔[J].物理学报,2011,60(12):66-73 Wu Z Q, Zhang Z H, Hao Y. Constrained bifurcations of the system with double-loop bilinear hysteresis[J]. Acta Physica Sinica,2011,60(12):66-73(in Chinese)
[4] 秦朝红.非线性动力学双参量奇异性方法及其工程应用[D].哈尔滨位:哈尔滨工业大学,2010 Qin C H. Singularity method for nonlinear dynamical analysis of systems with two parameters and its application in engineering[D]. Harbin:Harbin Institute of Technology, 2010(in Chinese)
[5] Mason J F, Piiroinen P T. Interactions between global and grazing bifurcations in an impacting system[J]. Chaos,2011,21(1):013113
[6] Thota P, Krauskopf B, Lowenberg M. Multi-parameter bifurcation study of shimmy oscillations in a dual-wheel aircraft nose landing gear[J]. Nonlinear Dynamics, 2012,70(2):1675-1688
[7] Li X F, Chu Y D, Zhang H. Fractal structures in a generalized square map with exponential terms[J]. Chinese Physics B, 2011,21(3):030203
[8] Ramírez-Ávila G M, Gallas J A C. How similar is the performance of the cubic and the piecewise-linear circuits of Chua?[J]. Physics Letters A, 2010,375(2):143-148
[9] 杨娟,卞保民,彭刚,等.随机信号双参数脉冲模型的分形特征[J].物理学报,2011,60(1):010508 Yang J, Bian B M, Peng G, et al. The fractal character of two-parameter pulse model for random signal[J]. Acta Physica Sinica,2011,60(1):010508(in Chinese)
[10] Gou X F, Zhu L Y, Chen D L. Bifurcation and chaos analysis of spur gear pair in two-parameter plane[J]. Nonlinear Dynamics, 2015,79(3):2225-2235
[11] 苟向锋,陈代林.双参变量下齿轮-转子系统扭转振动特性分析[J].工程力学,2014,31(11):211-217 Gou X F, Chen D L. Dynamic analysis on torsional vibration of gear-rotor system in two parameters plane[J]. Engineering Mechanics,2014,31(11):211-217(in Chinese)
[12] 谭涛亮,张尧,钟庆.交直流互联系统的多参数分岔分析[J].电力自动化设备,2012,32(2):23-28 Tan T L, Zhang Y, Zhong Q. Multi-parameter bifurcation analysis of AC/DC power system[J]. Electric Power Automation Equipment,2012,32(2):23-28(in Chinese)
[13] Zhang C, Bi Q S. On two-parameter bifurcation analysis of the periodic parameter-switching Lorenz oscillator[J]. Nonlinear Dynamics, 2015,81(1-2):577-583
[14] Luo G W, Lv X H, Shi Y Q. Vibro-impact dynamics of a two-degree-of freedom periodically-forced system with a clearance:diversity and parameter matching of periodic-impact motions[J]. International Journal of Non-Linear Mechanics,2014,65:173-195
[15] 吴淑花,孙毅,郝建红,等.耦合发电机系统的分岔和双参数特性[J].物理学报,2011,60(1):84-92 Wu S H, Sun Y, Hao J H, et al. Bifurcation and dual-parameter characteristic of the coupled dynamos system[J]. Acta Physica Sinica,2011,60(1):84-92(in Chinese)
[16] 刘秉正,彭建华.非线性动力学[M].北京:高等教育出版社,2004 Liu B Z, Peng J H. Nonlinear dynamics[M]. Beijing:Higher Education Press, 2004(in Chinese)