# Dynamic isoperimetry and the geometry of Lagrangian coherent structures

@article{Froyland2015DynamicIA, title={Dynamic isoperimetry and the geometry of Lagrangian coherent structures}, author={Gary Froyland}, journal={Nonlinearity}, year={2015}, volume={28}, pages={3587 - 3622} }

The study of transport and mixing processes in dynamical systems is particularly important for the analysis of mathematical models of physical systems. We propose a novel, direct geometric method to identify subsets of phase space that remain strongly coherent over a finite time duration. This new method is based on a dynamic extension of classical (static) isoperimetric problems; the latter are concerned with identifying submanifolds with the smallest boundary size relative to their volume…

## 60 Citations

Dynamic Isoperimetry on Weighted Manifolds

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Transport and mixing in dynamical systems are important mechanisms for many physical processes. We consider the detection of transport barriers using a recently developed geometric technique [1]: the…

A Dynamic Laplacian for Identifying Lagrangian Coherent Structures on Weighted Riemannian Manifolds

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The main results include generalised versions of the dynamic isoperimetric problem, the dynamic Laplacian, Cheeger’s inequality, and the Federer–Fleming theorem.

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- MathematicsBulletin of the Australian Mathematical Society
- 2018

This dissertation develops a data-driven approach for transport barrier detection, by extending and generalising dynamic isoperimetry to graphs and weighted Riemannian manifolds, and proves a dynamic Cheeger inequality for graphs.

A Geometric Heat-Flow Theory of Lagrangian Coherent Structures

- MathematicsJ. Nonlinear Sci.
- 2020

This approach facilitates the discovery of connections between some prominent methods for coherent structure detection: the dynamic isoperimetry methodology, the variational geometric approaches to elliptic LCSs, a class of graph Laplacian-based methods and the effective diffusivity framework used in physical oceanography.

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- 2015

A numerical method based on radial basis function collocation is presented and applied to a recent transfer operator construction that has been designed specifically for purely advective dynamics, leading to large speedups in the transfer operator analysis when this computation is costly.

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This work focuses on developing theory and methodologies for the analysis of material transport in stochastic fluid flows. In a first part, two dominant classes of techniques for extracting…

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It is argued that coherent sets are regions of maximal farness in terms of transport, hence they occur as extremal regions on a spanning structure of the state space under this semidistance - in fact, under any distance measure arising from the physical notion of transport.

Extraction and prediction of coherent patterns in incompressible flows through space–time Koopman analysis

- Computer Science
- 2017

Robust FEM-Based Extraction of Finite-Time Coherent Sets Using Scattered, Sparse, and Incomplete Trajectories

- MathematicsSIAM J. Appl. Dyn. Syst.
- 2018

Three FEM-based numerical methods are developed to efficiently approximate the dynamic Laplace operator, and a new dynamic isoperimetric problem using Dirichlet boundary conditions is introduced.

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