Duffing系统在双参数平面上的分岔演化过程

张艳龙; 王丽; 石建飞

doi:10.21656/1000-0887.380089

Duffing系统在双参数平面上的分岔演化过程

doi: 10.21656/1000-0887.380089

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

基金项目: 国家自然科学基金（11302092; 11362008）

详细信息

作者简介:
张艳龙（1981—），男，副教授，博士生，硕士生导师(E-mail: zhangyl@mail.lzjtu.cn)；王丽（1979—），女，副教授，硕士(通讯作者. E-mail: wangl@lzcu.edu.cn).

中图分类号: O322;O241.1
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出版历程
- 收稿日期: 2017-04-07
- 修回日期: 2017-04-26
- 刊出日期: 2018-03-15

Bifurcation Evolution of Duffing Systems on 2-Parameter Planes

¹School of Mechanical Engineering, Lanzhou Jiaotong University, Lanzhou 730070, P.R.China;²School of Mathematics, Lanzhou City University, Lanzhou 730070, P.R.China

Funds: The National Natural Science Foundation of China（11302092; 11362008）

摘要

摘要: 给出了参数空间上最大Lyapunov指数的计算方法，数值计算了Duffing系统在双参数平面上的最大Lyapunov指数.结合单参数最大Lyapunov指数、分岔图、相图以及时间历程图，讨论了Duffing系统在双参数平面上的分岔以及随系统控制参数变化的分岔演化过程.结果发现在双参数平面上系统发生叉式分岔，出现具有缺边现象的两个不同区域，该区域内系统对初值有较强的敏感性，存在两吸引子共存现象；系统运动经过周期跳跃曲线时振动幅值突然减小；系统外激励频率较小时常引起颤振运动.此外，在两个具有缺边现象的区域内，随刚度系数的不断增加，系统出现了倍周期分岔曲线环，而且倍周期分岔曲线环内不断嵌套新的倍周期分岔曲线环，导致系统最终经倍周期分岔序列进入混沌状态，随着控制参数的变化，系统在双参数平面上的动力学特性变得非常复杂.
- Duffing系统 /
- Lyapunov指数 /
- 双参数特性 /
- 分岔 /
- 周期跳跃
Abstract: The calculation method for the top Lyapunov exponents in the parameter space was given. The top Lyapunov exponents of Dufffing systems on 2-parameter planes were calculated with the numerical method. Combined with the single-parameter top Lyapunov exponents, the bifurcation diagrams, the phase diagrams and the time response diagrams, the bifurcation and the bifurcation evolution process of Duffing systems on the 2-parameter planes were discussed in view of the change of system parameters. The results show that 2 different regions with the phenomena of missing edges appear when the pitchfork bifurcation occurs. The system has strong sensitivity to initial values in the regions where 2 attractors coexist. The system vibration amplitude decreases suddenly when the system moves through the period jump curve. The system flutter motion often occurs when the excitation frequency is relatively small. In addition, when the stiffness coefficient increases, the period-doubling bifurcation curve cycles constantly exist and nest each other in the 2 regions with the phenomena of missing edges, which makes the system finally evolve into a chaotic state via the period-doubling bifurcation sequences. The dynamic properties of the system are very complex on 2-parameter planes with the change of control parameters.
- Duffing system /
- Lyapunov exponent /
- 2-parameter characteristic /
- bifurcation /
- periodic jump

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