  A classical particle in a confining potential gives rise to a Hamiltonian
conservative dynamical system with an uncountably infinite continuous energy
spectra, whereas the corresponding quantum particle exhibits countably infinite
discrete energy levels. We consider a class of nonlinear self-sustained
oscillators describing a classical active particle in a harmonic potential.
These nonlinear oscillators emerge in the low-memory regime of both
state-dependent time-delay systems as well as in non-Markovian stroboscopic
models of walking droplets. Using averaging techniques, we prove the existence
of a countably infinite number of asymptotically stable quantized orbits, i.e.
megastability, for this class of self-excited systems. The set of periodic
orbits consists of a sequence of nested limit-cycle attractors with quasilinear
increasing amplitude and alternating stability, yielding smooth basins of
attraction. By using the Lyapunov energy function, we estimate the energy
spectra of this megastable set of orbits, and perform numerical simulations to
confirm the mathematical analysis. Our formalism can be extended to
self-excited particles in general confining potentials, resulting in different
energy-frequency relations for these dynamical analogs of quantization.
