Pilot-Wave Dynamics: Using Dynamic Mode Decomposition to characterize Bifurcations, Routes to Chaos and Emergent Statistics

TL;DR

This paper uses Dynamic Mode Decomposition to analyze the bifurcations and chaos in a pilot-wave hydrodynamic system, revealing links to quantum mechanics and predicting particle statistics from wave dynamics.

Contribution

It introduces a data-driven DMD approach to characterize bifurcations and chaos in pilot-wave systems, connecting wave field evolution with droplet statistics.

Findings

Wave field undergoes Hopf bifurcations leading to chaos.

DMD provides a low-rank model of bifurcation structure.

Wave theory predicts particle statistics similar to quantum mechanics.

Abstract

We develop a data-driven characterization of the pilot-wave hydrodynamic system in which a bouncing droplet self-propels along the surface of a vibrating bath. We consider drop motion in a confined one-dimensional geometry, and apply the {\em Dynamic mode decomposition} (DMD) in order to characterize the evolution of the wave field as the bath's vibrational acceleration is increased progressively. DMD provides a regression framework for adaptively learning a best-fit linear dynamics model over snapshots of spatio-temporal data. The DMD characterization of the wave field yields a fresh perspective on the bouncing-droplet problem that forges valuable new links with the mathematical machinery of quantum mechanics. Moreover, it provides a low-rank characterization of the bifurcation structure of the pilot wave physics. Specifically, the analysis shows that as the vibrational acceleration is increased, the pilot-wave field undergoes a series of Hopf bifurcations that ultimately lead to a chaotic wave field. The established relation between the mean pilot-wave field and the droplet statistics allows us to characterize the evolution of the emergent statistics with increased vibrational forcing from the evolution of the pilot-wave field. We thus develop a numerical framework with the same basic structure as quantum mechanics, specifically a wave theory that predicts particle statistics.