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 $\mathbb{S}^{2}$.

翻译：在本文中,我们分析了找到最小维度($n)的问题,这样封闭式分析图/普通差分方程式就可以模拟图灵机超过$\mathb{R ⁇ n}美元,从而能够以强于扰动的方式模拟图灵机。我们显示一维封闭式分析图象足以强力模拟图灵机;但是封闭式分析普通差分方程式用于强力模拟图灵机的最小维度是两个,根据一些合理的假设。我们还显示,任何图灵机都可以用两维的 $C ⁇ infty}普通差异方程式模拟,在紧凑的球体上为$\mathb{S ⁇ 2}美元。