Durey Matthew, Milewski Paul A, Bush John W M
Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom.
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA.
Chaos. 2018 Sep;28(9):096108. doi: 10.1063/1.5030639.
A millimetric droplet may bounce and self-propel on the surface of a vertically vibrating bath, where its horizontal "walking" motion is induced by repeated impacts with its accompanying Faraday wave field. For ergodic long-time dynamics, we derive the relationship between the droplet's stationary statistical distribution and its mean wave field in a very general setting. We then focus on the case of a droplet subjected to a harmonic potential with its motion confined to a line. By analyzing the system's periodic states, we reveal a number of dynamical regimes, including those characterized by stationary bouncing droplets trapped by the harmonic potential, periodic quantized oscillations, chaotic motion and wavelike statistics, and periodic wave-trapped droplet motion that may persist even in the absence of a central force. We demonstrate that as the vibrational forcing is increased progressively, the periodic oscillations become chaotic via the Ruelle-Takens-Newhouse route. We rationalize the role of the local pilot-wave structure on the resulting droplet motion, which is akin to a random walk. We characterize the emergence of wavelike statistics influenced by the effective potential that is induced by the mean Faraday wave field.
一个毫米级的液滴可以在垂直振动的浴槽表面弹跳并自我推进，其水平“行走”运动是由与伴随的法拉第波场的反复碰撞引起的。对于遍历性的长时间动力学，我们在非常一般的情况下推导出液滴的静态统计分布与其平均波场之间的关系。然后我们关注一个受到简谐势作用且运动局限于一条直线上的液滴的情况。通过分析系统的周期状态，我们揭示了许多动力学状态，包括那些以被简谐势捕获的静止弹跳液滴、周期量子化振荡、混沌运动和波状统计为特征的状态，以及即使在没有中心力的情况下也可能持续的周期波捕获液滴运动。我们证明，随着振动强迫逐渐增加，周期振荡通过吕埃勒 - 塔肯斯 - 纽豪斯途径变为混沌。我们阐明了局部导波结构对由此产生的液滴运动的作用，这种运动类似于随机游走。我们描述了受平均法拉第波场诱导的有效势影响的波状统计的出现。
