A small droplet of silicon oil may bounce on a vertically vibrated bath of the same fluid, creating waves on the bath surface. As the vibration acceleration is increased, the bouncing is destabilised. A new stable regime forms in which the droplet 'walks' across the surface of the bath. The wave-droplet coupling allows for multi-droplet interactions, and trajectories including trefoils and butterflies.

From the linearised Navier-stokes equations, we make simplifications to allow for analysis of this system. This gives rise to a set of homogeneous linear periodic ODEs with jump conditions. We find necessary conditions for each steady state, but for an infinite dimensional system. Properties of the system allow for an a posteriori truncation, from which we solve using a nested Newton method combined with a shooting method. A simple iterative method is used to find continuous solution curves. Possible improvements to the numerical methods are discussed.