Transport in time-dependent dynamical systems: Finite-time coherent sets

Froyland, Gary; Santitissadeekorn, Naratip; Monahan, Adam H.

doi:10.1063/1.3502450

Cited by 191 publications

(336 citation statements)

References 38 publications

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“…This allows us to efficiently extract coherent sets of the underlying flow, similar in spirit to the approaches proposed in Hadjighasem et al (2016) and Banisch and Koltai (2017), who considered weighted networks, which are constructed based on using different metrics for determining the distance between two trajectories.…”

Section: Introductionmentioning

confidence: 99%

“…Such nearly decoupled subgraphs correspond to bundles of trajectories that are internally well connected but only loosely connected to other trajectories. This is indicative of coherent behavior (see also Hadjighasem et al, 2016). Instead of considering the eigenvalue problem Lw = λw, Shi and Malik (2000) propose to solve the equivalent generalized eigenvalue problem Lv = λDv.…”

Section: Spectral Graph Partitioningmentioning

confidence: 99%

“…In particular, we will describe how to extract coherent structures by solving a graph-partitioning problem, the balanced minimum cut problem as proposed by Shi and Malik (2000) (see also Hadjighasem et al, 2016).…”

Section: Network Analysismentioning

confidence: 99%

“…All these methods can deal with sparse and incomplete trajectory data and do respect the dynamics of the entire trajectories, not just the end points. While c-means clustering as used by Froyland and Padberg-Gehle (2015) is computationally inexpensive and works well in example systems (see also Allshouse and Peacock, 2015), spectral clustering approaches as in Hadjighasem et al (2016), Banisch andKoltai (2017), andSchlueter-Kuck andDabiri (2017) appear to be more robust, but require considerable computational effort.…”

Section: Introductionmentioning

confidence: 99%

“…Very recently, spatio-temporal clustering algorithms have been proven to be very effective for the extraction of coherent sets from sparse and possibly incomplete trajectory data (see, e.g., Froyland and Padberg-Gehle, 2015;Hadjighasem et al, 2016;Banisch and Koltai, 2017;Schlueter-Kuck and Dabiri, 2017). Here, distance measures between trajectories are used to define groups of trajectories that remain close and/or behave similarly in the time span under investigation.…”

Section: Introductionmentioning

confidence: 99%

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Network-based study of Lagrangian transport and mixing

Padberg‐Gehle

Schneide

2017

Nonlin. Processes Geophys.

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Abstract. Transport and mixing processes in fluid flows are crucially influenced by coherent structures and the characterization of these Lagrangian objects is a topic of intense current research. While established mathematical approaches such as variational methods or transfer-operatorbased schemes require full knowledge of the flow field or at least high-resolution trajectory data, this information may not be available in applications. Recently, different computational methods have been proposed to identify coherent behavior in flows directly from Lagrangian trajectory data, that is, numerical or measured time series of particle positions in a fluid flow. In this context, spatio-temporal clustering algorithms have been proven to be very effective for the extraction of coherent sets from sparse and possibly incomplete trajectory data. Inspired by these recent approaches, we consider an unweighted, undirected network, where Lagrangian particle trajectories serve as network nodes. A link is established between two nodes if the respective trajectories come close to each other at least once in the course of time. Classical graph concepts are then employed to analyze the resulting network. In particular, local network measures such as the node degree, the average degree of neighboring nodes, and the clustering coefficient serve as indicators of highly mixing regions, whereas spectral graph partitioning schemes allow us to extract coherent sets. The proposed methodology is very fast to run and we demonstrate its applicability in two geophysical flows -the Bickley jet as well as the Antarctic stratospheric polar vortex.

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Section: Introductionmentioning

confidence: 99%

Section: Spectral Graph Partitioningmentioning

confidence: 99%

Section: Network Analysismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Network-based study of Lagrangian transport and mixing

Padberg‐Gehle

Schneide

2017

Nonlin. Processes Geophys.

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Stochastic Stability of Lyapunov Exponents and Oseledets Splittings for Semi‐invertible Matrix Cocycles

Froyland

González-Tokman

Quas

2015

Comm Pure Appl Math

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We establish (i) stability of Lyapunov exponents and (ii) convergence in probability of Oseledets spaces for semi‐invertible matrix cocycles subjected to small random perturbations. The first part extends results of Ledrappier and Young to the semi‐invertible setting. The second part relies on the study of evolution of subspaces in the Grassmannian; the analysis developed here is based on higher‐dimensional Möbius transformations and is likely to be of wider interest. © 2015 Wiley Periodicals, Inc.

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Detecting the birth and death of finite‐time coherent sets

Froyland

Koltai

2023

Comm Pure Appl Math

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Finite‐time coherent sets (FTCSs) are distinguished regions of phase space that resist mixing with the surrounding space for some finite period of time; physical manifestations include eddies and vortices in the ocean and atmosphere, respectively. The boundaries of FTCSs are examples of Lagrangian coherent structures (LCSs). The selection of the time duration over which FTCS and LCS computations are made in practice is crucial to their success. If this time is longer than the lifetime of coherence of individual objects then existing methods will fail to detect the shorter‐lived coherence. It is of clear practical interest to determine the full lifetime of coherent objects, but in complicated practical situations, for example a field of ocean eddies with varying lifetimes, this is impossible with existing approaches. Moreover, determining the timing of emergence and destruction of coherent sets is of significant scientific interest. In this work we introduce new constructions to address these issues. The key components are an inflated dynamic Laplace operator and the concept of semi‐material FTCSs. We make strong mathematical connections between the inflated dynamic Laplacian and the standard dynamic Laplacian, showing that the latter arises as a limit of the former. The spectrum and eigenfunctions of the inflated dynamic Laplacian directly provide information on the number, lifetimes, and evolution of coherent sets.

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Transport in time-dependent dynamical systems: Finite-time coherent sets

Cited by 191 publications

References 38 publications

Network-based study of Lagrangian transport and mixing

Network-based study of Lagrangian transport and mixing

Stochastic Stability of Lyapunov Exponents and Oseledets Splittings for Semi‐invertible Matrix Cocycles

Detecting the birth and death of finite‐time coherent sets

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