解析一维映射和二维常微分方程可以鲁棒地模拟图灵机
Analytic one-dimensional maps and two-dimensional ordinary differential equations can robustly simulate Turing machines
摘要：
在本文中，我们分析的问题，找到最小尺寸$n$，使一个封闭形式的解析映射/常微分方程可以模拟图灵机在$\mathbb{R}^{n}$的方式，是强大的扰动。我们表明，一维封闭形式的解析映射是足够的鲁棒模拟图灵机，但封闭形式的解析常微分方程鲁棒模拟图灵机的最小维数是2，在一些合理的假设。我们还表明，任何图灵机可以模拟一个二维$C^{\infty}$常微分方程的紧凑球$\mathbb{S}^{2}$。
英文摘要：
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 $\mathbb{S}^{2}$.
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