  A bouncing droplet on a vibrated bath can couple to the waves it generates,
so that it becomes a propagative walker. Its propulsion at constant velocity
means that a balance exists between the permanent input of energy provided by
the vibration and the dissipation. Here we seek a simple theoretical
description of the resulting non-Hamiltonian dynamics with a walker immersed in
a harmonic potential well. We demonstrate that the interaction with the
recently emitted waves can be modeled by a Rayleigh-type friction. The Rayleigh
oscillator has well defined attractors. The convergence toward them and their
stability is investigated through an energetic approach and a linear stability
analysis. These theoretical results provide a description of the dynamics in
excellent agreement with the experimental data. It is thus a basic framework
for further investigations of wave-particle interactions when memory effects
are included.
