  We consider white noise perturbations of a nonlinear dynamical system in the
neighborhood of an unstable critical point with linearization given by a Jordan
block of full dimension. For the associated exit problem, we study the joint
limiting behavior of the exit location and exit time, in the vanishing noise
limit. The exit typically happens near one of two special deterministic points
associated with the eigendirection, and we obtain several more terms in the
expansion for the exit point. The leading correction term is deterministic and
logarithmic in the noise magnitude, while the random remainder satisfies a
scaling limit.
