  A droplet bouncing on the surface of a vertically vibrating liquid bath can
walk horizontally, guided by the waves it generates on each impact. This
results in a self-propelled classical particle-wave entity. By using a
one-dimensional theoretical pilot-wave model with a generalized wave form, we
investigate the dynamics of this particle-wave entity. We employ different
spatial wave forms to understand the role played by both wave oscillations and
spatial wave decay in the walking dynamics. We observe steady walking motion as
well as unsteady motions such as oscillating walking, self-trapped oscillations
and irregular walking. We explore the dynamical and statistical aspects of
irregular walking and show an equivalence between the droplet dynamics and the
Lorenz system, as well as making connections with the Langevin equation and
deterministic diffusion.
