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... and since chaotic systems have so far defeated mathematical modelling ...

Errm, no they haven't. Here's one I made earlier:

x → 4x(1–x)

That's the "logistic map". Here's another:

x' = s(y-x)

y' = x(r-z)-y

z' = xy-bz

That's the famous Lorentz system, which has chaotic solutions for some parameters. Chaotic systems are really easy to model. In fact, for continuous systems, as soon as you have enough variables and some nonlinearity you tend to get chaos.