Invariant Forms in Hybrid and Impact Systems and a Taming of Zeno
Invariant Forms in Hybrid and Impact Systems and a Taming of Zeno
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DOI：
10.1007/s00205-023-01844-1
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发表时间：
2021-01
影响因子：
2.5
通讯作者：
W. Clark;A. Bloch
中科院分区：
文献类型：
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作者：
W. Clark;A. Bloch
混合（和冲击）系统是经历连续和离散过渡的动态系统。在这项工作中，我们推导出必要和充分条件时，一个给定的微分形式是不变的，特别注意的情况下，存在不变的体积。特别注意的是连续动态拉格朗日和非完整约束的影响系统。保体积动力系统的一个著名结果是庞加莱递归。为了是循环的，轨迹需要存在很长一段时间，这可以在连续时间系统中通过例如紧凑性来控制。对于混合系统，可能会出现一种额外的机制来打破长期存在：芝诺（在有限的时间内有无限多个离散的转变）。我们证明了一个光滑不变体积的存在严重抑制芝诺行为;混合系统的“边界身份属性”沿着与不变体积形式几乎没有芝诺轨迹（虽然芝诺轨迹仍然可以存在）。这导致了许多台球（例如经典点，滚动盘和滚动球）是独立于紧凑桌面形状的递归。
Hybrid (and impact) systems are dynamical systems experiencing both continuous and discrete transitions. In this work, we derive necessary and sufficient conditions for when a given differential form is invariant, with special attention paid to the case of the existence of invariant volumes. Particular attention is given to impact systems where the continuous dynamics are Lagrangian and subject to nonholonomic constraints. A celebrated result for volume-preserving dynamical systems is Poincaré recurrence. In order to be recurrent, trajectories need to exist for long periods of time, which can be controlled in continuous-time systems through e.g. compactness. For hybrid systems, an additional mechanism can occur which breaks long-time existence: Zeno (infinitely many discrete transitions in a finite amount of time). We demonstrate that the existence of a smooth invariant volume severely inhibits Zeno behavior; hybrid systems with the “boundary identity property” along with an invariant volume-form have almost no Zeno trajectories (although Zeno trajectories can still exist). This leads to the result that many billiards (e.g. the classical point, the rolling disk, and the rolling ball) are recurrent independent on the shape of the compact table-top.