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 are computable. Our findings suggest that the global stability of a system plays 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。我们构造了一个可计算的$C^{\infty}$系统$x^{\prime}=f(x)$，该系统具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。具体而言，当$f$受到微小扰动时，该平衡点及其关联吸引域的不可计算性均保持不变。这表明局部稳定性本身不足以确保吸引域的可计算性。然而，我们同时也证明了结构稳定（全局稳定）平面系统所对应的吸引域是可计算的。我们的研究结果表明，系统的全局稳定性在决定其吸引域可计算性方面起着关键作用。