A dynamic Laplacian for identifying Lagrangian coherent structures on weighted Riemannian manifolds

TL;DR

This paper extends the dynamic Laplacian framework to identify transport barriers in complex, non-volume-preserving, and curved dynamical systems, providing new theoretical results and computational methods for analyzing phase space structures.

Contribution

It generalizes the dynamic isoperimetric problem and Laplacian to broader settings, including non-volume-preserving dynamics and curved phase spaces, with new theoretical and computational tools.

Findings

Generalized dynamic Laplacian and isoperimetric problem

Established Cheeger's inequality and Federer-Fleming theorem in new settings

Numerical examples demonstrating the approach

Abstract

Transport and mixing in dynamical systems are important properties for many physical, chemical, biological, and engineering processes. The detection of transport barriers for dynamics with general time dependence is a difficult, but important problem, because such barriers control how rapidly different parts of phase space (which might correspond to different chemical or biological agents) interact. The key factor is the growth of interfaces that partition phase space into separate regions. The paper [Froyland, Nonlinearity 2015] introduced the notion of \textit{dynamic isoperimetry}: the study of sets with persistently small boundary size (the interface) relative to enclosed volume, when evolved by the dynamics. Sets with this minimal boundary size to volume ratio were identified as level sets of dominant eigenfunctions of a \textit{dynamic Laplace operator}. In this present work we extend the results of [Froyland, Nonlinearity 2015] to the situation where the dynamics (i) is not necessarily volume-preserving, (ii) acts on initial agent concentrations different from uniform concentrations, and (iii) occurs on a possibly curved phase space. Our main results include generalised versions of the dynamic isoperimetric problem, the dynamic Laplacian, Cheeger's inequality, and the Federer-Fleming theorem. We illustrate the computational approach with some simple numerical examples.