Megastable quantization in generalized pilot-wave hydrodynamics

A classical particle in a harmonic potential gives rise to a continuous energy spectra, whereas the corresponding quantum particle exhibits countably infinite quantized energy levels. In recent years, classical non-Markovian wave-particle entities that materialize as walking droplets have been shown to exhibit various hydrodynamic quantum analogs, including quantization in a harmonic potential by displaying few coexisting limit cycle orbits. By considering a truncated-memory stroboscopic pilot-wave model of the system in the low dissipation regime, we obtain a classical harmonic oscillator perturbed by oscillatory non-conservative forces that displays countably infinite coexisting limit-cycle states, also known as \emph{megastability}. Using averaging techniques in the low-memory regime, we derive analytical approximations of the orbital radii, orbital frequency and Lyapunov energy function of this megastable spectrum, and further show average energy conservation along these quantized states. Our formalism extends to a general class of self-excited oscillators and can be used to construct megastable spectrum with different energy-frequency relations.