磁场中受垂直激励非线性单摆的动力学分析  

Dynamic Analysis of Vertical Excited Pendulum in Magnetic Field

作　　者：慕云田 李险峰[1] MU Yuntian;LI Xianfeng(School of Mathematics and Physics,Lanzhou Jiaotong University,Lanzhou 730070,China)

机构地区：[1]兰州交通大学数理学院,兰州730070

出　　处：《兰州交通大学学报》2024年第3期116-126,共11页Journal of Lanzhou Jiaotong University

基　　金：国家自然科学基金(11962011);甘肃省自然科学基金(20JR10RA233,22JR5RA348)。

摘　　要：在结合能量采集等生活实际应用领域的基础上,构建了一个在磁场中的单摆模型,并深入分析了其动力学行为,旨在为工程摆设备的工作性能参数的优化选择提供理论支持。首先通过势能函数导出了势阱曲线,并探讨了势阱和势垒对系统动力特性的影响。然后采用Melnikov函数方法,不仅求解了系统的同宿轨迹和产生Smale意义上混沌的临界条件,还揭示了单摆系统振幅与系统频率之间的关系,并提出了计算同宿分岔曲线的数值方法,成功绘制了分岔曲线。最后通过综合运用单参数分岔、Lyapunov指数、相轨迹图、庞加莱截面图以及双参数平面上的分岔图等分析工具,展示了摆动系统演进至混沌的两种路径:倍周期分岔和鞍结分岔。此外,还揭示了这类单摆系统的全局运动规律,如倍周期分岔、吸引子倍增、鞍结分岔等,进一步明确了相关参数对工程系统工作性能的影响及其作用机制。Based on the practical application of energy harvesting,a mathematical pendulum model in a magnetic field is constructed,and its dynamic behavior is analyzed in depth,The purpose of this study is to provide theoretical sup-port for the optimal selection of working performance parameters of engineering pendulum equipment.Firstly,the potential well curve is derived from the potential energy function,and the influence of the potential well and barrier on the dynamic characteristics of the system is discussed.Then,using the Melnikov function method,this study not only solves the homoclinic trajectory of the system and the critical conditions for the generation of chaos in the Smale sense,but also reveals the relationship between the amplitude and the frequency of the simple pendulum system,and proposes a numerical method to calculate the homoclinic bifurcation curve,and successfully draw the bifurcation curve.Finally,by using the analytical tools such as single-parameter bifurcation,Lyapunov exponent,phase trajectory diagram,Poincare cross section diagram and bifurcation diagram in the two-parameter plane,two paths of the evolu-tion of the oscillating system to chaos are shown:period-doubling bifurcation and saddle-junction bifurcation.In addi-tion,the global motion laws of this kind of simple pendulum system,such as period doubling bifurcation,attractor doubling,saddle junction bifurcation,etc.,are also revealed,and the influence of related parameters on the working performance of the engineering system and the action mechanism are further clarified.

关 键 词：磁摆 分岔 混沌 二维参数空间 

分 类 号：O322[理学—一般力学与力学基础]