Computational Method for Phase Space Transport with Applications to Lobe Dynamics and Rate of Escape

TL;DR

This paper introduces a new computational method for analyzing phase space transport in chaotic systems, focusing on intersection points and areas between curves, with applications in fluid mechanics and ship dynamics.

Contribution

It presents a novel theory and algorithm for accurately computing intersections and enclosed areas between curves in phase space, including nontransverse intersections, implemented in open-source software.

Findings

Effective computation of intersection points and areas in phase space.

Application to fluid mechanics and ship dynamics.

Open-source software Lober facilitates these computations.

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 are 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 periodic orbit around saddles. Both of these frameworks require computation with curves represented by millions of points-computing intersection points between these curves and area bounded by the segments of these curves-for quantifying the transport and escape rate. We present a theory for computing these intersection points and the area bounded between the segments of these curves based on a classification of the intersection points using equivalence class. We also present an alternate theory for curves with nontransverse intersections and a method to increase the density of points on the curves for locating the intersection points accurately.The numerical implementation of the theory presented herein is available as an open source software called Lober. We used this package to demonstrate the application of the theory to lobe dynamics that arises in fluid mechanics, and rate of escape from a potential well that arises in ship dynamics.