Oscillator in the simplest physics sense (a passive oscillatory system e.g. spring-and-mass), or practically speaking (a driven system with feedback)? Or more specifically still, a combination of the two (a periodic or even ~sinusoidal oscillator)?
A complete description to some very, very basic level, I think you'll have some trouble expressing, but if we say it's a given that atoms and molecules exist at around room temperature, then a basic chemical oscillator operating on a concentration gradient of reactants might be sufficient.
Not sure offhand if there are simpler systems than that; perhaps some gas molecules have interesting enough dynamics that say a heat flux, or light, or excess kinetic energy (a molecular or particle beam?) is enough to induce oscillation. We could of course point to plasmas, but shall we include atomic transitions as instances of oscillation or not?
There are molecular vibration modes that closely resemble a spring-and-mass system (quantum harmonic oscillator, QHO), wherein the energy levels are quantized, but the transition energy between adjacent states is approximately constant -- equivalent to the fixed resonant frequency of the classical harmonic oscillator. This is in contrast to the more sporadic modes of a bare atom for example, where similar dynamics might apply (with suitable stimulus (i.e., from its spectrum), we can increment between electron orbital levels), but not all states can be transitioned between, in a single event, due to spin selection rules. Whereas the QHO has a "ladder" state that allows free exchange of potential energy at the same frequency.
From the more abstract side -- a linear ODE isn't sufficient description, as we necessarily have limited sources and sinks of energy/power. We can get nearly perpetual oscillators if we remove all influences; a lone planet in circular orbit of a lone star, pretty closely obeys such an equation, over time scales of billions of years (for suitable definition of "lone"); but not for all time, and eventually there will be significant exchange of energy with surrounding systems. (If nothing else, even for ideal rigid bodies orbiting in isolation from any others, gravitational waves will decay the orbit over heat-death time scales. Or maybe it's less than that? I forget.) Particularly for a driven system, we must include a logistic function that, given an input, limits maximum amplitude or energy exchange; and extraction, given an output. All real systems are nonlinear, having such a limit.
More generally, we might specify an oscillator with a limit cycle, given some assumed energy input, exchange, and output. The 2nd order term might even be dominant (as in the above case), but other terms are necessary in a real system.
The complexity then, is finding a way to express these terms, which is both simple enough to be useful, and real enough to be meaningful. I'm afraid I don't know enough about DEs anymore to help further with such a description. Anyway, the plan would be to start from a high level like this, and also from a low level (in whatever medium you like, be it fields, particles or combinations thereof), and meet in the middle with a complete description of an implementation/realization of that system.
Assuming I've interpreted your meaning and intent correctly, of course!
Tim