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.

翻译：本文研究了动力系统 $x^{\prime}=f(x)$ 的稳定性与其吸引盆可计算性之间的关系。我们提出了一个可计算的 $C^{\infty}$ 系统 $x^{\prime}=f(x)$，该系统具有一个可计算且稳定的平衡点，但其吸引盆在 $f$ 的某个邻域内是鲁棒不可计算的，即当 $f$ 受到微小扰动时，该平衡点及其对应吸引盆的不可计算性均保持不变。这表明，仅靠稳定平衡点附近的局部稳定性不足以保证其吸引盆的可计算性。然而，我们也证明了定义在紧集上的结构稳定——全局稳定（鲁棒）——平面系统，其对应的吸引盆是可计算的。我们的研究结果表明，系统的全局稳定性与定义域的紧致性在决定其吸引盆的可计算性方面起着关键作用。