  Consider a two-dimensional continuous-time dynamical system, with an
attracting fixed point $S$. If the deterministic dynamics are perturbed by
white noise (random perturbations) of strength $\epsilon$, the system state
will eventually leave the domain of attraction $\Omega$ of $S$. We analyse the
case when, as $\epsilon\to0$, the exit location on the boundary
$\partial\Omega$ is increasingly concentrated near a saddle point $H$ of the
deterministic dynamics. We show that the asymptotic form of the exit location
distribution on $\partial\Omega$ is generically non-Gaussian and asymmetric,
and classify the possible limiting distributions. A key role is played by a
parameter $\mu$, equal to the ratio $|\lambda_s(H)|/\lambda_u(H)$ of the stable
and unstable eigenvalues of the linearized deterministic flow at $H$. If
$\mu<1$ then the exit location distribution is generically asymptotic as
$\epsilon\to0$ to a Weibull distribution with shape parameter $2/\mu$, on the
$O(\epsilon^{\mu/2})$ length scale near $H$. If $\mu>1$ it is generically
asymptotic to a distribution on the $O(\epsilon^{1/2})$ length scale, whose
moments we compute. The asymmetry of the asymptotic exit location distribution
is attributable to the generic presence of a `classically forbidden' region: a
wedge-shaped subset of $\Omega$ with $H$ as vertex, which is reached from $S$,
in the $\epsilon\to0$ limit, only via `bent' (non-smooth) fluctuational paths
that first pass through the vicinity of $H$. We deduce from the presence of
this forbidden region that the classical Eyring formula for the
small-$\epsilon$ exponential asymptotics of the mean first exit time is
generically inapplicable.

Adaptation, Noise, and Self-Organizing Systems;Chaotic Dynamics;Adaptation and Self-Organizing Systems;Chaotic Dynamics