  We consider the dynamics of a droplet on a vibrating fluid bath. This
hydrodynamic quantum analog system is shown to elicit the canonical behavior of
damped-driven systems, including a period doubling route to chaos. By
approximating the system as a compositional map between the gain and loss
dynamics, the underlying nonlinear dynamics can be shown to be driven by energy
balances in the systems. The gain-loss iterative mapping is similar to a normal
form encoding for the pattern forming instabilities generated in such
spatially-extended system. Similar to mode-locked lasers and rotating
detonation engines, the underlying bifurcations persist for general forms of
the loss and gain, both of which admit explicit representations in our
approximation. Moreover, the resulting geometrical description of the
particle-wave interaction completely characterizes the instabilities observed
in experiments.
