  Bouncing droplets on a vibrating fluid bath can exhibit wave-particle
behavior, such as being propelled by interacting with its own wave field. These
droplets seem to walk across the bath, and thus are dubbed walkers. Experiments
have shown that walkers can exhibit exotic dynamical behavior indicative of
chaos. While the integro-differential models developed for these systems agree
well with the experiments, they are difficult to analyze mathematically. In
recent years, simpler discrete dynamical models have been derived and studied
numerically. The numerical simulations of these models show evidence of exotic
dynamics such as period doubling bifurcations, Neimark--Sacker (N--S)
bifurcations, and even chaos. For example, in [Gilet, PRE 2014], based on
simulations Gilet conjectured the existence of a supercritical N-S bifurcation
as the damping factor in his one-dimensional path model. We prove Gilet's
conjecture and more; in fact, both supercritical and subcritical (N-S)
bifurcations are produced by separately varying the damping factor and
wave-particle coupling for all eigenmode shapes. Then we compare our
theoretical results with some previous and new numerical simulations, and find
complete qualitative agreement. Furthermore, evidence of chaos is shown by
numerically studying a global bifurcation.
