  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 wave-particle
entity 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.
