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A classical wave-particle entity (WPE) can materialize as a millimeter-sizeddroplet walking horizontally on the free surface of a vertically vibratingliquid bath. This WPE comprises a particle (droplet) that shapes itsenvironment by locally exciting decaying standing waves, which in turn guidesthe particle motion. At high amplitude of bath vibrations, theparticle-generated waves decay very slowly in time and the particle motion isinfluenced by the history of waves along its trajectory. In this high-memoryregime, WPEs exhibit hydrodynamic quantum analogs where quantum-like statisticsarise from underlying chaotic dynamics. Exploration of WPE dynamics in the veryhigh-memory regime requires solving an integro-differential equation of motion.By using an idealized one-dimensional WPE model where the particle generatessinusoidal waves, we show that in the limit of infinite memory, the systemdynamics reduce to a $3$D nonlinear system of ordinary differential equations(ODEs) known as the diffusionless Lorenz equations (DLEs). We use ouralgebraically simple ODE system to explore in detail, theoretically andnumerically, the rich set of periodic and chaotic dynamical behaviors exhibitedby the WPE in the parameter space. Specifically, we link the geometry anddynamics in phase-space of the DLE system to the dynamical and statisticalfeatures of WPE motion, paving a way to understand hydrodynamic quantum analogsusing phase-space attractors. Our system also provides an alternateinterpretation of an attractor-driven particle, i.e.~an active particle drivenby internal state-space variables of the DLE system. Hence, our results mightalso provide new insights in modeling active particle locomotion.

Infinite-memory classical wave-particle entities, attractor-driven active particles and the diffusionless Lorenz equations