Chaos, Solitons and Fractals 186 (2024) 115253 Available online 23 July 2024 0960-0779/© 2024 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Contents lists available at ScienceDirect Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos Quantum-like behavior of an active particle in a double-well potential Rahil N. Valani a,c ,∗ , Álvaro G. López b a School of Computer and Mathematical Sciences, University of Adelaide, Adelaide, 5005, South Australia, Australia b Nonlinear Dynamics, Chaos and Complex Systems Group. Departamento de Física, Universidad Rey Juan Carlos, Tulipán s/n, Móstoles, 28933, Madrid, Spain c Rudolf Peierls Centre for Theoretical Physics, Parks Road, University of Oxford, Oxford, OX1 3PU, United Kingdom ARTICLE INFO MSC: 0000 1111 Keywords: Wave–particle duality Hydrodynamic quantum analogs Walking droplets Lorenz system Active particles Time-delay systems ABSTRACT A macroscopic, self-propelled wave–particle entity (WPE) that emerges as a walking droplet on the surface of a vibrating liquid bath exhibits several hydrodynamic quantum analogs. We explore the rich dynamical and quantum-like features emerging in a model of an idealized one-dimensional WPE in a double-well potential. The integro-differential equation of motion for the WPE transforms to a Lorenz-like system, which we explore in detail. We observe the analog of quantized eigenstates as discrete limit cycles that arise by varying the width of the double-well potential, and also in the form of multistability with coexisting limit cycles. These states show narrow as well as wide energy level splitting. Tunneling-like behavior is also observed where the WPE erratically transitions between the two wells of the double-well potential. We rationalize this phenomena in terms of crisis-induced intermittency. Further, we discover a fractal structure in the escape time distribution of the particle from a well based on initial conditions, indicating unpredictability of this tunneling-like intermittent behavior at all scales. The chaotic intermittent dynamics lead to wave-like emergent features in the probability distribution of particle’s position that show qualitative similarity with its quantum counterpart. Lastly, rich dynamical features are also observed such as a period doubling route to chaos as well as self-similar periodic islands in the chaotic parameter set. 1. Introduction Quantization, wave-like statistics and tunneling are usually taken to be exclusive features of the microscopic quantum realm. However, in recent years, a hydrodynamic system of millimeter-sized walking droplets [1,2] has demonstrated that classical systems with memory effects can exhibit several hydrodynamic quantum analogs. In this hydrodynamic system, a millimetric droplet bounces periodically while walking horizontally on the free surface of a vertically vibrating bath of the same liquid. Each bounce of the droplet generates a localized and slowly decaying standing wave on the liquid surface. The droplet interacts with these self-generated waves on subsequent bounces result- ing in horizontal motion. This gives rise to self-propelled walking and superwalking [3] droplets. Three key physical properties to note about this system are: (i) The droplet and its underlying wave coexist as a wave–particle entity (WPE); without the droplet the underlying waves decay completely. (ii) The droplet is active in the sense of an active particle and active matter [4], since the droplet locally extracts energy from the vibrating liquid bath and converts it into directed motion. (iii) There is memory in the system. Since the waves generated by the droplet decay very slowly in time, the motion of the droplet is not only influenced by the wave it generated on the most recent bounce, ∗ Corresponding author at: Rudolf Peierls Centre for Theoretical Physics, Parks Road, University of Oxford, Oxford, OX1 3PU, United Kingdom. E-mail address: rahil.valani@physics.ox.ac.uk (R.N. Valani). but also by the waves it generated in the distant past. Summarizing, the walking-droplet system constitutes a locally active system with memory effects embedded in its dynamics. This allows the particle to amplify energy fluctuations from its environment, converting them into coherent motion and breaking fundamental symmetries [5]. Many of the hydrodynamic quantum analogs exhibited by this active WPE, in both experiments as well as simulations, typically take place in the high-memory regime and/or when the motion of the WPE is spatially confined. For example, confinement of a WPE in external potentials with a central force results in quantized orbits [6–11]. By confining a WPE in a two-dimensional harmonic potential, a discrete set of orbits are observed such as circles, ovals, lemniscates and tre- foils [6,9]. Here, the dynamic constraint imposed on the particle by its guiding wave field results in quantum-like eigenstates emerging from wave-memory mediated interactions. Hydrodynamic analog of quan- tum tunneling has also been shown for a WPE. In experiments, the WPE interacting with a submerged barrier typically gets reflected from the barrier. Eddi et al. [12] showed that occasionally, the interactions of the WPE with the barrier can lead to droplet tunneling across the barrier. Thus, the complex interaction of the particle with its underlying wave field results in unpredictable tunneling. In the high-memory regime https://doi.org/10.1016/j.chaos.2024.115253 Received 10 January 2024; Received in revised form 17 June 2024; Accepted 6 July 2024