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 - 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 $x^{\prime}=f(x)$ 的稳定性与其吸引域可计算性之间的关系。我们构建了一个可计算的 $C^{\infty}$ 系统 $x^{\prime}=f(x)$，该系统具有一个可计算且稳定的平衡点，但在 $f$ 的邻域内，其吸引域是鲁棒不可计算的——即当 $f$ 受到轻微扰动时，该平衡点及其关联吸引域的不可计算性均得以保持。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域的可计算性。然而，我们同时证明，在紧集上定义的、结构稳定（即全局稳定）的平面系统所关联的吸引域是可计算的。我们的研究结果表明，系统的全局稳定性及定义域的紧致性在判定其吸引域的可计算性中起着关键作用。