A Dynamical Systems Approach for Most Probable Escape Paths over Periodic Boundaries
Abstract
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 EulerLagrange system, which we call the River. Using the Maslov index we identify a subset of the River which is comprised of local minimizers. The OnsagerMachlup (OM) functional, which is treated as a perturbation of the FriedlinWentzell 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 (socalled IVDP). Through MonteCarlo simulations, we show that the prediction provided by OMselected MPEP matches closely the escape hatch chosen by noisy trajectories at a certain level of small noise.
 Publication:

arXiv eprints
 Pub Date:
 February 2023
 DOI:
 10.48550/arXiv.2302.00758
 arXiv:
 arXiv:2302.00758
 Bibcode:
 2023arXiv230200758F
 Keywords:

 Mathematics  Dynamical Systems;
 65K10;
 11Y16;
 60G17;
 37J50;
 34C45
 EPrint:
 27 pages, 14 figures