It's fascinating that whereas the solution of the *quartic* oscillator (restoring force $\propto r^3$) problem can be expressed as the distance-from-origin of a point moving with unit speed along a lemniscate $r=\sqrt{\cos(2\theta)}$ (made a silly mistake there at first - fixed it now), the solution for an ordinary *quadratic* oscillator (restoring force $\propto r$) can likewise be expressed as the distance-from-origin of a point moving with unit speed along a *fattened* quasi-lemniscate $r=\cos\theta$ ... or two congruent circles just fitting within the unit circle.

Unlike as the case of the quartic oscillator, which can infact be elemenarily realised, a *prototype* realisation being transverse oscillations of a mass at the centre of a taught elastic string, I can't offhand think of how a *hektic* oscillator (restoring force $\propto r^5$) might simply be physically realised. Would that have its quasi-lemniscate?

I suppose it *must exist* ... it's the

*solubility*of it, really, that raises the questions. I think the lobes would be rather

*teardrop*-shaped ... on the grounds that as the exponent increases, the scenario becomes more like that of the particle simply

*bouncing*off 'sheer walls' of force - so the lemniscate would tend more & more to be two increasingly flattened 'blobs' each on the end of an increasingly thin 'stalk' lying diametrically opposed to the other.

I think it might be one of those situations in which you have two figures, one for each of two scenarios one of which is one step further further along in a progression, that have a simple & beautiful relationship to each other; but in which when you try to extend it indefinitely it just starts getting kind of ... *ugly*. Quaternions - octonions, etc; exponentiation - tetration, etc ... it seems to happen with such *regularity*. And sometimes you just *so want* the progression to be extensible indefinitely!

**Update**

See answer.