Computational method for phase space transport with applications to lobe dynamics and rate of escape

Abstract

Lobe dynamics and escape from a potential well are general frameworks introduced to study phase space transport in chaotic dynamical systems.While the former approach studies how regions of phase space get transported by reducing the flow to a two-dimensional map, the latter approach studies the phase space structures that lead to critical events by crossing certain barriers. Lobe dynamics describes global transport in terms of lobes, parcels of phase space bounded by stable and unstable invariant manifolds associated to hyperbolic fixed points of the system. Escape from a potential well describes how the critical events occur and quantifies the rate of escape using the flux across the barriers. Both of these frameworks require computation of curves, intersection points, and the area bounded by the curves. We present a theory for classification of intersection points to compute the area bounded between the segments of the curves. This involves the partition of the intersection points into equivalence classes to apply the discrete form of Green’s theorem. We present numerical implementation of the theory, and an alternate method for curves with nontransverse intersections is also presented along with a method to insert points in the curve for densification.