Analyzing when noisy trajectories, in the two dimensional plane, of a
stochastic dynamical system exit the basin of attraction of a fixed point is
specifically challenging when a periodic orbit forms the boundary of the basin
of attraction. Our contention is that there is a distinguished Most Probable
Escape Path (MPEP) crossing the periodic orbit which acts as a guide for noisy
escaping paths in the case of small noise slightly away from the limit of
vanishing noise. It is well known that, before exiting, noisy trajectories will
tend to cycle around the periodic orbit as the noise vanishes, but we observe
that the escaping paths are stubbornly resistant to cycling as soon as the
noise becomes at all significant. Using a geometric dynamical systems approach,
we isolate a subset of the unstable manifold of the fixed point in the
Euler-Lagrange system, which we call the River. Using the Maslov index we
identify a subset of the River which is comprised of local minimizers. The
Onsager-Machlup (OM) functional, which is treated as a perturbation of the
Friedlin-Wentzell functional, provides a selection mechanism to pick out a
specific MPEP. Much of the paper is focused on the system obtained by reversing
the van der Pol Equations in time (so-called IVDP). Through Monte-Carlo
simulations, we show that the prediction provided by OM-selected MPEP matches
closely the escape hatch chosen by noisy trajectories at a certain level of
small noise.Comment: 28 pages, 15 figure