Not sure if this is appropriate to Math Overflow, but I think there's some way to make this precise, even if I'm not sure how to do it myself.


Say I have a nasty ODE, nonlinear, maybe extremely singular. It showed up naturally mathematically (I'm actually thinking of Painleve VI, which comes from isomonodromy representations) but I've got a bit of a physicist inside me, so here's the question. Can I construct, in every case, a physical system modeled by this equation? Maybe even just some weird system of coupled harmonic oscillators, something. There are a few physical systems whose models are well understood, and I'm basically asking if there's a construction that takes an ODE and constructs some combination of these systems that it controls the dynamics of.


Any input would be helpful, even if it's just "No." though in that case, a reason would be nice.



Does every ODE comes from something in physics?
It is possible to solve a large class of ODEs by means of analog computers. Each of the pieces of the differential equation corresponds to an electronic component and if you wire them up the right way you get a circuit described by the ODE. [Wikipedia](https://en.wikipedia.org/wiki/Analog_computer) has lots of information on the subject and a link like [this](http://www.analogmuseum.org/library/handson.pdf) one gives explicit examples of circuits. It's not hard to build circuits for things like the [Lorenz equation](https://en.wikipedia.org/wiki/Lorenz_attractor) and see a nice Lorenz attractors on an [oscilloscope display](https://www.youtube.com/watch?v=6HWfdRZo1I4).


