The Domain of Attraction of the Desired Path in Vector-Field-Guided Path Following

In the vector-field-guided path-following problem, a sufficiently smooth vector field is designed such that its integral curves converge to and move along a 1-D geometric desired path. The existence of singular points where the vector field vanishes creates a topological obstruction to global convergence to the desired path and some associated topological analysis has been conducted in our previous work. In this article, we further show that the domain of attraction of the desired path, which is a compact asymptotically stable 1-D embedded submanifold of an
$n$
-dimensional ambient manifold
$\mathcal {M}$
, is homeomorphic to
$\mathbb{R}^{n-1} \times \mathbb{S}^{1}$
, and not just homotopy equivalent to
$\mathbb{S}^{1}$
.