A Dynamic Laplacian for Identifying Lagrangian Coherent Structures on Weighted Riemannian Manifolds
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A Dynamic Laplacian for Identifying Lagrangian Coherent Structures on Weighted Riemannian Manifolds Gary Froyland1 · Eric Kwok1 Received: 4 October 2016 / Accepted: 23 May 2017 © Springer Science+Business Media, LLC 2017
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 28(10):3587–3622, 2015) introduced the notion of 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 dynamic Laplace operator. In this present work we extend the results of Froyland (Nonlinearity 28(10):3587–3622, 2015) to the situation where the dynamics (1) is not necessarily volume preserving, (2) acts on initial agent concentrations different from uniform concentrations, and (3) 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. Keywords Lagrangian coherent structure · Dynamic Laplace operator · Weighted Riemannian manifold · Transfer operator · Finite-time coherent set · Dynamic isoperimetric problem
Communicated by Clancy Rowley and Ioannis Kevrekidis.
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Gary Froyland [email protected] School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia
123
J Nonlinear Sci
Mathematics Subject Classification 37C60 · 53C21 · 53A10 · 58J50
1 Introduction The mathematics of transport in nonlinear dynamical systems has received considerable attention for more than two decades, driven in part by applications in fluid dynamics, atmospheric and ocean dynamics, molecular dynamics, granular flow, and other areas. We refer the reader to Ottino (1989), Rom-Kedar et al. (1990), Meiss (1992), Aref (2002), Wiggins (2005) for reviews of transport and transport-related phenomena. Early attempts to characterise transport barriers in fluid dynamics include time-dependent invariant manifolds (such as lobe dynamics Rom-Kedar et al. 1990) and finite-time Lyapunov exponents (Pierrehumbert 1991; Pierrehumbert and Yang 1993; Doerner et al. 1999; Haller 2002; Shadden et al. 2005). More recently, in twodimensional area-preserving flows, Haller and Beron-Vera (2013) proposed finding closed curves whose time-averaged length is stationary under small perturbations; this aim is closest in spirit1 to the predecessor
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