Two balls in one dimension with gravity.
Two balls in one dimension with gravity.
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一维中的两个球具有重力。
DOI：
10.1103/physreva.42.742
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发表时间：
1990
期刊：
影响因子：
--
通讯作者：
Cannizzo
中科院分区：
文献类型：
--
作者：
Whelan;Goodings;Cannizzo
一项研究报告了一个具有两个自由度的简单动力系统，该系统由于碰撞而具有不连续性。它由两个点质量或球组成，它们被限制在恒定重力场中的地板上方的一维中移动。假设所有碰撞都是弹性的。当上部质量与下部质量的比率 r 小于 1 时，运动几乎到处都是混沌的。另一方面，当 r > 1 时，运动表现出典型的 Kolmogorov-Arnol’d-Moser 行为，准周期轨迹和混沌轨迹在相空间中共存。结果表明，对于由 r n 表示的特定质量比值，靠近地板的 n 次快速球与球碰撞的序列具有反转两个球速度的净效果。这个演示导致了庞加莱截面稳定和不稳定不动点族的识别，这在很大程度上决定了地图的整体结构。通过洛伦兹的方法，已经针对许多质量比值和多种轨迹计算了最大李雅普诺夫指数λ 1 。对于混沌轨迹，我们发现 λ 1 作为 r 函数的图在对应于速度反转碰撞序列的值 r n 处具有局部最小值。这被认为是由于当 r= r n 时，混沌轨迹位于相空间的许多孤立区域中，而当 r 与任何 r n 不同时，混沌区域合并形成全局混沌的单个区域。
A study is reported of a simple dynamical system with two degrees of freedom having discontinuities due to collisions. It consists of two point masses or balls constrained to move in one dimension above a floor in a constant gravitational field. All collisions are assumed to be elastic. When the ratio r of the upper mass to the lower mass is less than unity the motion is chaotic almost everywhere. On the other hand, when r> 1 the motion shows typical Kolmogorov-Arnol’d-Moser behavior with quasiperiodic and chaotic trajectories coexisting in the phase space. It is shown that for particular values of the mass ratio, denoted by r n, a sequence of n rapid ball-ball collisions close to the floor has the net effect of reversing the velocities of both balls. This demonstration leads to the identification of families of stable and unstable fixed points of the Poincaré section, which to a considerable extent determine the overall structure of the map. By means of a method due to Lorenz, the largest Lyapunov exponent λ 1 has been calculated for many values of the mass ratio and for a variety of trajectories. For chaotic trajectories, a plot of λ 1 as a function of r is found to have local minima at the values r n corresponding to velocity-reversing collision sequences. This is thought to result from the fact that when r= r n the chaotic trajectories lie in many isolated regions of the phase space, whereas when r is different from any of the r n, the chaotic regions merge to form a single region of global chaos.