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In this paper, we analyze the problem of finding the minimum dimension $n$such that a closed-form analytic map/ordinary differential equation cansimulate a Turing machine over $\mathbb{R}^{n}$ in a way that is robust toperturbations. We show that one-dimensional closed-form analytic maps aresufficient to robustly simulate Turing machines; but the minimum dimension forthe closed-form analytic ordinary differential equations to robustly simulateTuring machines is two, under some reasonable assumptions. We also show thatany Turing machine can be simulated by a two-dimensional $C^{\infty}$ ordinarydifferential equation on the compact sphere $\mathbb{S}^{2}$.

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