The prototype harmonic oscillator in physics discussions is a spring
How about a pendulum ? Or does the externality of gravity complicate things?
We could do some interesting categorization with respect to the decay mechanism (or conversely, gain when applicable). Note that simple rotation decays over time, resulting in a reduced frequency. That's a very nonlinear sort of behavior compared to SHM. Depending on the loss mechanism, the decay might be exponential, or double-exponential or something else (compare constant frictional resistance vs. friction proportional to angular velocity*). Or it might be increasing, with a hyperbolic decay curve -- as for an orbital system (frequency ~ 1/energy) decaying under gravitational radiation.
*Well... there was a kernel of something more interesting in my mind, but it doesn't sound very special when written out, does it? Simple resistance is torque proportional to angular velocity, which gives a simple pole (exponential decay), the usual case. But there are mechanisms which are strongly dependent on rotation rate, or frequency (in quantum terms, E ~ ν, but the emission / coupling rate matters too, and I don't know offhand any figures for that), or say temperature (radiation ~ T^4).
Say we stick a magnet or battery to the rotor, so it's emitting EM waves at the rotation frequency; at most frequencies, it will be a short dipole so loss varies strongly with frequency. (Ah, but does it vary proportionally, or square law, or more? I don't recall offhand**.)
But these are more easily collected by specifying an amplitude- or rate-dependent function, and observe the special cases for loss = 0 (perpetual) and loss ~ rate (constant resistance), then solve for whatever the resulting decay curve is (or infer the opposite way just as well).
**Well, induced voltage (relative to a non-rotating reference frame) is proportional to rate, and, what, a short dipole has radiation resistance as 1/λ^2? Which will be the series equivalent resistance of the lossy capacitor, so...
V ~ ω
I ~ V / Xc ~ V ω ~ ω^2
λ ~ 1/ω
R ~ 1 / λ^2 ~ ω^2
P ~ I^2 R
P ~ ω^6 ??
One class is that any convex shape is analogous to a sphere. (There is a better mathematical term for this, but I can't recall it right now.)
Gaussian curvature..?
There is a third one, too: the smaller the amplitude compared to the energies and distances involved in an oscillator, the closer it seems to match a harmonic oscillator. However, I myself do not know why this is, or even if it is just a practical result of a noisy universe (i.e., any non-harmonic components drowned in noise in real life), so I consider it more or less just a rule of thumb.
In general that's just linearization around a point; in this case, a small cycle around phase space. If the system is continuous, it approximates a SHO orbit.
I suppose for certain dynamic systems, we can employ transformations to convert limit cycles into limit points, and map non-circular trajectories onto unit circles (thus accounting for harmonic distortion). The existence of such a mapping, on a given system, might suggest its generalized-SHO-ness.
It gets even more bamboozling when thinking about numerical oscillators. What would mean energy conservation for a numerical oscillator, or momentum, or seeking minimum energy equilibrium, or inertia? All these are very important and inescapable in a physical oscillator, yet they don't make much sense, and they seem totally arbitrary for a numerical oscillator.
NCOs are abstracted in a different direction: an iterated transformation (discrete-time) on a discrete state vector (i.e. bits in memory). We could indeed apply some implementation (whether transistors or otherwise), assign an energy or power cost to the states, or transitions therebetween, and model it the same way -- just with a lot of extra steps.
More tantalizing, I suppose, might be applying such maps as above -- suppose we implement an NCO using "lossless" digital logic, at such a clock rate that the loss tangent of the capacitances is sufficiently negligible, and using an LC oscillator (and whatever applicable phase-shift networks) to afford the various clock phases required for operation. Then we could have a digital description of a continuous oscillating system, which exhibits rather complex behavior (various period-multiplication effects as the counters roll over from time to time) but overall is nonetheless, not just coherent with, but indeed energetically coupled to, our reference oscillator.
The standard implementation today is very much more mundane; since our information-theoretic efficiency is piss (something like ~10^6 times more energy loss per bit transformation), it's basically a class A amplifier: all power dissipation all the time. And with all that wasted power, it really doesn't matter what happens. You have the space for a lot of complicated -- but thermodynamically irrelevant -- functions.

Tim