标题： 两个物体在杆上摆动的规则与混沌运动 显示英文标题

标题： Regular and Chaotic Motion of Two Bodies Swinging on a Rod

摘要： 我们研究两个物体在杆上摆动的规则和混沌动力学，这与其他所有机械类比不同：根据初始条件，其振荡可能会很快结束，而原因不是阻尼力或能量损失。 我们使用各种工具来分析运动，例如 准周期和混沌情况下的庞加莱截面。 我们通过不同的方法计算李雅普诺夫特征指数，包括有限时间李雅普诺夫指数分析。 我们的计算表明，除了在边缘情况下观察到准周期振荡时，最大李雅普诺夫指数总是正的。 显示英文摘要

摘要： We investigate regular and chaotic dynamics of Two Bodies Swinging on a Rod, which differs from all the other mechanical analogies: depending on initial conditions, its oscillation could end very quickly and the reason is not a drag force or energy loss. We use various tools to analyze motion, such as Poincar\'e section for quasi-periodic and chaotic cases. We calculate Lyapunov characteristic exponent by different methods including Finite Time Lyapunov Exponent analysis. Our calculations show that the maximal Lyapunov exponent is always positive except in the marginal cases when one observes quasi-periodic oscillations.