Bifurcation Evolution of Duffing Systems on 2-Parameter Planes
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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.
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关键词:
- 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.-
Key words:
- Duffing system /
- Lyapunov exponent /
- 2-parameter characteristic /
- bifurcation /
- periodic jump
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[1] SHAW S W, HOLMES P J. A periodically forced piecewise linear oscillator[J]. Journal of Sound and Vibration,1983,90(1): 129-155. [2] 夏南, 孟光. 非线性系统周期强迫不平衡响应的稳定性分析[J]. 力学学报, 2001,33(1): 128-133.(XIA Nan, MENG Guang. The analysis on the stability of responses of strong nonlinear system subjected to periodic unbalance force[J].Acta Mechanica Sinica,2001,33(1): 128-133.(in Chinese)) [3] NORDMARK A B. Non-periodic motion caused by grazing incidence in an impact oscillator[J]. Journal of Sound and Vibration,1991,145(2): 279-297. [4] 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. [5] GOU Xiangfeng, ZHU Lingyun, CHEN Dailin. Bifurcation and chaos analysis of spur gear pair in two-parameter plane[J]. Nonlinear Dynamics,2015,79(3): 2225-2235. [6] 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): 010507.(WU Shuhua, SUN Yi, HAO Jianhong, et al. Bifurcation and dual-parameter characteristic of the coupled dynamos system[J]. Acta Physica Sinica,2011,60(1): 010507.(in Chinese)) [7] 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. [8] 杨娟, 卞保明, 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 Juan, BIAN Baoming, PENG Gang, et al. The fractal character of two-parameter pulse model for random signal[J]. Acta Physica Sinica,2011,60(1): 010508.(in Chinese)) [9] DAI Guowei. Two global several-parameter bifurcation theorems and their applications[J]. Journal of Mathematical Analysis and Applications,2016,433(2): 749-761. [10] VAN LOI N. On two-parameter global bifurcation of periodic solutions to a class of differential variational inequalities[J]. Nonlinear Analysis: Theory, Methods & Applications,2015,122: 83-99. [11] SHAW P K, JANAKI M S, IYENGAR A N S, et al. Antiperiodic oscillations in a forced Duffing oscillator[J]. Chaos, Solitons & Fractals,2015,78: 256-266. [12] JOHANNESSEN K. The Duffing oscillator with damping[J]. European Journal of Physics, 2015,36(6): 065020. DOI: 10.1088/0143-0807/36/6/065020. [13] CHU Jifeng, WANG Feng. Prevalence of stable periodic solutions for Duffing equations[J]. Journal of Differential Equations,2016,260(11): 7800-7820. [14] RUSINEK R, WEREMCZUK A, KECIK K, et al. Dynamics of a time delayed Duffing oscillator[J]. International Journal of Non-Linear Mechanics,2014,65: 98-106. [15] 武娟, 许勇. 加性二值噪声激励下Duffing系统的随机分岔[J]. 应用数学和力学, 2015,36(6): 593-599.(WU Juan, XU Yong. Stochastic bifurcations in a Duffing system driven by additive dichotomous noises[J].Applied Mathematics and Mechanics,2015,36(6): 593-599.(in Chinese)) [16] 张莹, 都琳, 岳晓乐, 等. 随机参数作用下参激双势阱Duffing系统的随机动力学行为分析[J]. 应用数学和力学, 2016,37(11): 1198-1207.(ZHANG Ying, DU Lin, YUE Xiaole, et al. Stochastic nonlinear dynamics analysis of double-well Duffing systems under random parametric excitations[J]. Applied Mathematics and Mechanics,2016,37(11): 1198-1207.(in Chinese)) [17] XIE Dan, XU Min, DAI Honghua, et al. Observation and evolution of chaos for a cantilever plate in supersonic flow[J]. Journal of Fluids and Structures,2014,50: 271-291.
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