  A classical wave-particle entity (WPE) can materialize as a millimeter-sized
droplet walking horizontally on the free surface of a vertically vibrating
liquid bath. This WPE comprises a particle (droplet) that shapes its
environment by locally exciting decaying standing waves, which in turn guides
the particle motion. At high amplitude of bath vibrations, the
particle-generated waves decay very slowly in time and the particle motion is
influenced by the history of waves along its trajectory. In this high-memory
regime, WPEs exhibit hydrodynamic quantum analogs where quantum-like statistics
arise from underlying chaotic dynamics. Exploration of WPE dynamics in the very
high-memory regime requires solving an integro-differential equation of motion.
By using an idealized one-dimensional WPE model where the particle generates
sinusoidal waves, we show that in the limit of infinite memory, the system
dynamics reduce to a $3$D nonlinear system of ordinary differential equations
(ODEs) known as the diffusionless Lorenz equations (DLEs). We use our
algebraically simple ODE system to explore in detail, theoretically and
numerically, the rich set of periodic and chaotic dynamical behaviors exhibited
by the WPE in the parameter space. Specifically, we link the geometry and
dynamics in phase-space of the DLE system to the dynamical and statistical
features of WPE motion, paving a way to understand hydrodynamic quantum analogs
using phase-space attractors. Our system also provides an alternate
interpretation of an attractor-driven particle, i.e.~an active particle driven
by internal state-space variables of the DLE system. Hence, our results might
also provide new insights in modeling active particle locomotion.
