Analytic one-dimensional maps and two-dimensional ordinary differential equations can robustly simulate Turing machines

In this paper, we analyze the problem of finding the minimum dimension n such that an analytic map/ordinary differential equation over R-n can simulate a Turing machine in a way that is robust to perturbations. We show that one-dimensional analytic maps are sufficient to robustly simulate Turing machines; but the minimum dimension for the analytic ordinary differential equations to robustly simulate Turing machines is two, under some reasonable assumptions. We also show that any Turing machine can be simulated by a two-dimensional C-infinity ordinary differential equation on the compact sphere S-2.