  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.
