Coherent sets for nonautonomous dynamical systems

TL;DR

This paper introduces a mathematical framework and algorithms for identifying and tracking coherent sets in nonautonomous dynamical systems, extending autonomous methods to handle time-dependent behaviors and slow mixing regions.

Contribution

It extends existing autonomous approaches to nonautonomous systems, enabling the detection of time-dependent coherent sets that disperse slowly and remain coherent over time.

Findings

New algorithms successfully identify coherent sets in various examples.

The methods work for both discrete and continuous time systems.

Coherent sets are shown to be important in understanding nonautonomous dynamics.

Abstract

We describe a mathematical formalism and numerical algorithms for identifying and tracking slowly mixing objects in nonautonomous dynamical systems. In the autonomous setting, such objects are variously known as almost-invariant sets, metastable sets, persistent patterns, or strange eigenmodes, and have proved to be important in a variety of applications. In this current work, we explain how to extend existing autonomous approaches to the nonautonomous setting. We call the new time-dependent slowly mixing objects coherent sets as they represent regions of phase space that disperse very slowly and remain coherent. The new methods are illustrated via detailed examples in both discrete and continuous time.