  Consider a parameter dependent vector field on either Euclidean space or a
compact Riemannian manifold. Suppose that it possesses a parameter dependent
initial condition and a parameter dependent stable hyperbolic equilibrium
point. It is valuable to determine the set of parameter values, which we call
the recovery set, whose corresponding initial conditions lie within the region
of attraction of the corresponding stable equilibrium point. A boundary
parameter value is a parameter value whose corresponding initial condition lies
in the boundary of the region of attraction of the corresponding stable
equilibrium point. Prior algorithms numerically estimated the recovery set by
estimating its boundary via computation of boundary parameter values. The
primary purpose of this work is to provide theoretical justification for those
algorithms for a large class of parameter dependent vector fields. This
includes proving that, for these vector fields, the boundary of the recovery
set consists of boundary parameter values, and that the properties exploited by
the algorithms to compute these desired boundary parameters will be satisfied.
The main technical result which these proofs rely on is establishing that the
region of attraction boundary varies continuously in an appropriate sense with
respect to small variation in parameter value for this class of vector fields.
Hence, the majority of this work is devoted to proving this result, which may
be of independent interest. The proof of continuity proceeds by proving that,
for this class of vector fields, the region of attraction permits a
decomposition into a union of the stable manifolds of the equilibrium points
and periodic orbits it contains, and this decomposition persists under small
perturbations to the vector field.
