  In this paper, we analyze the problem of finding the minimum dimension $n$
such that a closed-form analytic map/ordinary differential equation can
simulate a Turing machine over $\mathbb{R}^{n}$ in a way that is robust to
perturbations. We show that one-dimensional closed-form analytic maps are
sufficient to robustly simulate Turing machines; but the minimum dimension for
the closed-form 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 $C^{\infty}$ ordinary
differential equation on the compact sphere $\mathbb{S}^{2}$.
