Recent experiments show that quasi-one-dimensional lattices of self-propelled droplets exhibit collective instabilities in the form of out-of-phase oscillations and solitary-like waves. This hydrodynamic lattice is driven by the external forcing of a vertically vibrating fluid bath, which invokes a field of subcritical Faraday waves on the bath surface, mediating the spatio-temporal droplet coupling. By modelling the droplet lattice as a memory-endowed system with spatially non-local coupling, we herein rationalize the form and onset of instability in this new class of dynamical oscillator. We identify the memory-driven instability of the lattice as a function of the number of droplets, and determine equispaced lattice configurations precluded by geometrical constraints. Each memory-driven instability is then classified as either a super- or subcritical Hopf bifurcation via a systematic weakly nonlinear analysis, rationalizing experimental observations. We further discover a previously unreported symmetry-breaking instability, manifest as an oscillatory–rotary motion of the lattice. Numerical simulations support our findings and prompt further investigations of this nonlinear dynamical system.

中文翻译：

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流体动力主动晶格的集体振动

最近的实验表明，自推进液滴的准一维晶格以异相振荡和孤立波的形式表现出集体不稳定性。这种流体动力晶格是由垂直振动的流体浴的外力驱动的，它在浴表面上调用亚临界法拉第波场，介导时空液滴耦合。通过将液滴晶格建模为具有空间非局部耦合的记忆赋予系统，我们在此合理化了这种新型动态振荡器中不稳定性的形式和开始。我们将晶格的内存驱动不稳定性确定为液滴数量的函数，并确定几何约束排除的等距晶格配置。然后通过系统的弱非线性分析将每个记忆驱动的不稳定性分类为超临界或亚临界 Hopf 分岔，从而使实验观察合理化。我们进一步发现了以前未报告的对称破坏不稳定性，表现为晶格的振荡旋转运动。数值模拟支持我们的发现，并促使对该非线性动力系统的进一步研究。