单摆混沌现象的研究

Researches of chaos phenomena due to simple pendulum

利用MATLAB软件强大的数值计算功能,使用龙格库塔法严格求解无阻尼单摆和有阻尼单摆的运动方程,研究单摆摆长、阻尼系数和初始摆角对单摆运动的影响.研究结果表明:单摆的周期与初始摆角有关,单摆周期随初始摆角的增大而增大;当两种单摆的参数取值处于某些范围时,均能出现混沌现象;阻尼单摆在正阻尼条件下演化出随机吸引子,在负阻尼条件下演化出随机排斥子.通过计算不同参数值的单摆方程,发现单摆的运动对初始摆角、阻尼系数有很强的依赖性.最后提出了一种衡量混沌系统敏感性的量化指标——敏感系数,计算结果表明,初始摆角、单摆摆长、阻尼系数均能影响单摆的敏感系数.

Impact of length, damping coefficient and initial swing angle of simple pendulum on movement were studied by strictly solving equations of motion of the undamped simple pendulum and the damped pendulum respectively, which was supported by MATLAB software and Runge-Kutta method. The results showed that the initial swing angle could enhance the period of simple pendulum. Appropriate values of parameters could lead to chaos in two kinds of simple pendulums. For the positive damping case, random attractor appeared and for the negative damping case, random repeller appeared. Solving the movement equations with different parameter values, we found that movement depended on the initial swing angle and damping coefficient strongly. Finally, a quantifiable indicators weighing the degree of chaotic system＇s sensitivity was put forward, named sensitivity coefficient. Computation indicated that length, damping coefficient and initial swing angle had an effect on the sensitivity coefficient of simple pendulum.