An analytic framework for identifying finite-time coherent sets in time-dependent dynamical systems

TL;DR

This paper introduces a new analytic transfer operator framework for identifying finite-time coherent sets in time-dependent dynamical systems, accounting for diffusion effects and applicable to continuous Markov processes.

Contribution

It develops a novel transfer operator method to compute finite-time coherent sets, extending invariant set analysis to nonautonomous systems with diffusion considerations.

Findings

The new operator enables finite-time coherent set detection.

Spectral gap and singular vector regularity scale with noise amplitude.

Method applies to general Markov processes on continuous spaces.

Abstract

The study of transport and mixing processes in dynamical systems is particularly important for the analysis of mathematical models of physical systems. Barriers to transport, which mitigate mixing, are currently the subject of intense study. In the autonomous setting, the use of transfer operators (Perron-Frobenius operators) to identify invariant and almost-invariant sets has been particularly successful. In the nonautonomous (time-dependent) setting, \emph{coherent sets}, a time-parameterised family of minimally dispersive sets, are a natural extension of almost-invariant sets. The present work introduces a new analytic transfer operator construction that enables the calculation of \emph{finite-time} coherent sets (sets are that minimally dispersive over a finite time interval). This new construction also elucidates the role of diffusion in the calculation and we show how properties such as the spectral gap and the regularity of singular vectors scale with noise amplitude. The construction can also be applied to general Markov processes on continuous state space.