一句话：构造局部稳定却使吸引域鲁棒不可计算的可计算系统，并证明紧集上结构稳定全局稳定平面系统的吸引域可计算。
论文研究动力系统 x'=f(x) 的稳定性与其吸引域可计算性之间的关系，核心问题是局部稳定能否保证吸引域可计算。作者构造了一个可计算的 C∞ 系统：它拥有可计算且稳定的平衡点，但其吸引域在 f 的邻域内是鲁棒不可计算的，即平衡点与吸引域的不可计算性在轻微扰动下仍保持，说明局部稳定性不足以保证可计算性。另一方面，作者证明定义在紧集上的结构稳定—全局稳定（鲁棒）平面系统，其吸引域是可计算的。结果表明系统的全局稳定性与定义域的紧致性对吸引域可计算性起关键作用。
In this paper, we examine the relationship between the stability of the dynamical system $x^{\prime}=f(x)$ and the computability of its basins of attraction. We present a computable $C^{\infty}$ system $x^{\prime}=f(x)$ that possesses a computable and stable equilibrium point, yet whose basin of attraction is robustly non-computable in a neighborhood of $f$ in the sense that both the equilibrium point and the non-computability of its associated basin of attraction persist when $f$ is slightly perturbed. This indicates that local stability near a stable equilibrium point alone is insufficient to guarantee the computability of its basin of attraction. However, we also demonstrate that the basins of attraction associated with a structurally stable - globally stable (robust) - planar system defined on a compact set are computable. Our findings suggest that the global stability of a system and the compactness of the domain play a pivotal role in determining the computability of its basins of attraction.
@article{paperbot2749,
title = {Robust non-computability of dynamical systems and computability of robust dynamical systems},
author = {Daniel S. Graça and Ning Zhong},
journal = {Logical Methods in Computer Science},
volume = {Volume 20, Issue 2},
year = {2024},
doi = {10.46298/lmcs-20(2:19)2024}
}