  Stochastic systems are used to model a variety of phenomena in which noise
plays an essential role. In these models, one potential goal is to determine if
noise can induce transitions between states, and if so, to calculate the most
probable escape path from an attractor. In the small noise limit, the
Freidlin-Wentzell theory of large deviations provides a variational framework
to calculate these paths. This work focuses on using large deviation theory to
calculate such paths for stochastic gradient systems with non-gradient
perturbations. While for gradient systems the most probable escape paths
consist of time-reversed heteroclinic orbits, for general systems it can be a
challenging calculation. By applying Melnikov theory to the resulting
Euler-Lagrange equations recast in Hamiltonian form, we determine a condition
for when the optimal escape path is the heteroclinic orbit for the perturbed
system. We provide a numerical example to illustrate how the computed most
probable escape path compares with the theoretical result.
