Transport in Transitory Dynamical Systems

TL;DR

This paper introduces a new framework for quantifying transport in transitory dynamical systems, focusing on the Hamiltonian case and leveraging invariant manifolds to analyze fluid flows and particle accelerators.

Contribution

It presents a novel method to measure transport using heteroclinic orbit actions in transitory systems, simplifying computations by exploiting autonomous segments.

Findings

Effective quantification of transport in transitory systems.

Comparison with finite-time Lyapunov exponents and adiabatic theory.

Applicable to fluid flows and particle accelerators.

Abstract

We introduce the concept of a "transitory" dynamical system---one whose time-dependence is confined to a compact interval---and show how to quantify transport between two-dimensional Lagrangian coherent structures for the Hamiltonian case. This requires knowing only the "action" of relevant heteroclinic orbits at the intersection of invariant manifolds of "forward" and "backward" hyperbolic orbits. These manifolds can be easily computed by leveraging the autonomous nature of the vector fields on either side of the time-dependent transition. As illustrative examples we consider a two-dimensional fluid flow in a rotating double-gyre configuration and a simple one-and-a-half degree of freedom model of a resonant particle accelerator. We compare our results to those obtained using finite-time Lyapunov exponents and to adiabatic theory, discussing the benefits and limitations of each method.