The basic assumption of perturbation theory is that the frequency of the motion changes with the amplitude and as the amplitude increases the frequency shifts across resonances that change the motion.
In the case of the harmonic oscillator, the frequency doesn't change so this doesn't apply at all. Instead of working from that as the ε=0 limit, you work from an integrable system like
https://en.wikipedia.org/wiki/Toda_lattice
which models the changing frequency w/ amplitude and breaks the symmetry of the system.
The case of orbital motion in three dimensions (classically the same problem for celestial mechanics and an electron in a hydrogen atom) is also problematic because the frequency of the three motions (i) around the star, (ii) in-and-out away from the star, and (iii) up-and-down out of the equatorial plane are all the same so it is a degenerate case and takes consideration of how exactly the symmetry breaks.
I'd contrast that to quantum theory where the perturbation theory "just works" for the simple cases like the harmonic oscillator and orbital motion. (That's how you develop the theory of spectral lines for the hydrogen atom, as the electromagnetic field is modeled as a perturbation.)
Note: I studied chaos theory for my PhD a long time ago. Unfortunately very few people have been trained in this field and gotten professorships since 1970 or so, so the field is stillborn and people are still learning from old textbooks. People praise V.I. Arnold's book on classical mechanics but it was written before people knew about the integrability of the Toda Lattice and circa 1995 you could make it through a graduate level class on classical dynamics and not learn the right way to think about the "perturbed harmonic oscillator".