However, there are a number of differential equations where to simulate it requires exponential complexity w.r.t. to the given degree of accuracy or amount of "time" to simulate it for. For instance, pretty much any chaotic problem.
Agreed! There's a reason that non-linear equations are regarded as much harder than linear ones, and I don't mean to gloss over it. However, possibly just because I am a mathematician and not an engineer or hacker, I still find the distinction between "solution that is hard to find" and "no solution" meaningful and important. (It is the difference between "we can't predict the weather" and "there is no such thing as weather"!)
The classic example is Navier–Stokes equations of viscous fluid flow. True story, one of my college professors handed out a homework assignment that no one could solve. Far as I could tell the boundary conditions described journal bearing that wasn't stable. Such things exist.
I think other very hard to deal with are equations where the solution diverges exponentially due to small changes in boundary conditions. Orbital mechanics is like that.
Not sure but I can see a situation where the set of equations that describe the solution form a series that never converges to a regular pattern. (irrational series? as irrational numbers but with equations) I think quantum mechanics is like that (Don't quote me)
You can build a physical Turing machine and describe its behavior using Newton's equations of motion, then do a change of variable t -> 1/t. The value at 0 gives you the result of the computation. Q.E.D
That's … unsatisfying. I mean that it's unsatisfying in a theoretical sense (because it gives me no theoretical feeling for the Turing-machine-to-differential-equation transformation), but, if I really wanted to be picky, I could point out that (a) nothing in the definition of a Turing machine guarantees its physical realisability (what if its number of states is bigger than the power set of the number of elementary particles in the universe, or, similarly, if it writes on an unbounded amount of tape?), and (b) Newton's equations of motion are only approximations to, not exact descriptions of, the physical universe.
Is there any more-or-less explicit recipe that says "given a description of a Turing machine (as a 7-tuple, say https://en.wikipedia.org/wiki/Turing_machine#Formal_definiti...), here is a (possibly unmanageably huge) differential equation such that …"—well, I don't even really know what. Your answer suggests that I might ask that, say, the solution $y$ to the differential equation where $y(1)$ somehow encodes a given initial state of the tape is such that $y(0)$ somehow encodes the final state of the tape (with the understanding that $y$ is not defined at $0$ if the machine doesn't halt on the corresponding input).
> Fine, fine, I just think it's an elegant way to say it :)
You're absolutely right that it's an elegant and compelling argument for the plausibility of the claim; I was just looking for the rigour behind it (even a statement, if not a proof). Your second reference is exactly the sort of thing that I had in mind; thanks!
However, there are a number of differential equations where to simulate it requires exponential complexity w.r.t. to the given degree of accuracy or amount of "time" to simulate it for. For instance, pretty much any chaotic problem.