Dynamical systems often exhibit the emergence of long-lived coherent sets, which are regions in state space that keep their geometric integrity to a high extent and thus play an important role in transport. In this article, we provide a method for extracting coherent sets from possibly sparse Lagrangian trajectory data. Our method can be seen as an extension of diffusion maps to trajectory space, and it allows us to construct “dynamical coordinates,” which reveal the intrinsic low-dimensional organization of the data with respect to transport. The only a priori knowledge about the dynamics that we require is a locally valid notion of distance, which renders our method highly suitable for automated data analysis. We show convergence of our method to the analytic transfer operator framework of coherence in the infinite data limit and illustrate its potential on several two- and three-dimensional examples as well as real world data.
References
1.
G.
Froyland
, S.
Lloyd
, and A.
Quas
, “Coherent structures and isolated spectrum for Perron–Frobenius cocycles
,” Ergodic Theory Dyn. Syst.
30
, 729
–756
(2010
).2.
G.
Froyland
, S.
Lloyd
, and N.
Santitissadeekorn
, “Coherent sets for nonautonomous dynamical systems
,” Physica D
239
, 1527
–1541
(2010
).3.
G.
Froyland
, N.
Santitissadeekorn
, and A.
Monahan
, “Transport in time-dependent dynamical systems: Finite-time coherent sets
,” Chaos
20
, 043116
(2010
).4.
I. I.
Rypina
, M. G.
Brown
, F. J.
Beron-Vera
, H.
Koçak
, M. J.
Olascoaga
, and I. A.
Udovydchenkov
, “On the Lagrangian dynamics of atmospheric zonal jets and the permeability of the stratospheric polar vortex
,” J. Atmos. Sci.
64
, 3595
–3610
(2007
).5.
A.-M.
Treguier
, O.
Boebel
, B.
Barnier
, and G.
Madec
, “Agulhas eddy fluxes in a 1/6 degrees Atlantic model
,” Deep Sea Res. Part II
50
, 251
–280
(2003
).6.
M.
Dellnitz
, G.
Froyland
, C.
Horenkamp
, K.
Padberg-Gehle
, and A. S.
Gupta
, “Seasonal variability of the subpolar gyres in the Southern Ocean: A numerical investigation based on transfer operators
,” Nonlinear Processes Geophys.
16
, 655
–664
(2009
).7.
G.
Froyland
, C.
Horenkamp
, V.
Rossi
, and E.
van Sebille
, “Studying an Agulhas ring's long-term pathway and decay with finite-time coherent sets
,” Chaos
25
, 083119
(2015
).8.
K.
Padberg
, T.
Hauff
, F.
Jenko
, and O.
Junge
, “Lagrangian structures and transport in turbulent magnetized plasmas
,” New J. Phys.
9
, 400
(2007
).9.
M. R.
Allshouse
and J.-L.
Thiffeault
, “Detecting coherent structures using braids
,” Physica D
241
, 95
–105
(2012
).10.
M.
Budišić
and I.
Mezić
, “Geometry of the ergodic quotient reveals coherent structures in flows
,” Physica D
241
, 1255
–1269
(2012
).11.
V.
Rom-Kedar
and S.
Wiggins
, “Transport in two-dimensional maps
,” Arch. Ration. Mech. Anal.
109
, 239
–298
(1990
).12.
G.
Haller
, “Finding finite-time invariant manifolds in two-dimensional velocity fields
,” Chaos
10
, 99
–108
(2000
).13.
G.
Haller
and F. J.
Beron-Vera
, “Geodesic theory of transport barriers in two-dimensional flows
,” Physica D
241
, 1680
–1702
(2012
).14.
G.
Haller
, “Distinguished material surfaces and coherent structures in three-dimensional fluid flows
,” Physica D
149
, 248
–277
(2001
).15.
T.
Ma
and E. M.
Bollt
, “Differential geometry perspective of shape coherence and curvature evolution by finite-time nonhyperbolic splitting
,” SIAM J. Appl. Dyn. Syst.
13
, 1106
–1136
(2014
).16.
S.
Balasuriya
, G.
Froyland
, and N.
Santitissadeekorn
, “Absolute flux optimising curves of flows on a surface
,” J. Math. Anal. Appl.
409
, 119
–139
(2014
).17.
M.
Dellnitz
and O.
Junge
, “On the approximation of complicated dynamical behavior
,” SIAM J. Numer. Anal.
36
, 491
–515
(1999
).18.
G.
Froyland
and M.
Dellnitz
, “Detecting and locating near-optimal almost-invariant sets and cycles
,” SIAM J. Sci. Comput.
24
, 1839
–1863
(2003
).19.
G.
Froyland
and K.
Padberg
, “Almost-invariant sets and invariant manifolds - Connecting probabilistic and geometric descriptions of coherent structures in flows
,” Physica D
238
, 1507
–1523
(2009
).20.
G.
Froyland
and K.
Padberg-Gehle
, “Finite-time entropy: A probabilistic approach for measuring nonlinear stretching
,” Physica D
241
, 1612
–1628
(2012
).21.
G.
Froyland
, “Dynamic isoperimetry and the geometry of Lagrangian coherent structures
,” Nonlinearity
28
, 3587
–3622
(2015
).22.
M. R.
Allshouse
and T.
Peacock
, “Lagrangian based methods for coherent structure detection
,” Chaos
25
, 097617
(2015
).23.
E.
Ser-Giacomi
, V.
Rossi
, C.
López
, and E.
Hernández-García
, “Flow networks: A characterization of geophysical fluid transport
,” Chaos
25
, 036404
(2015
).24.
G.
Froyland
and K.
Padberg-Gehle
, “A rough-and-ready cluster-based approach for extracting finite-time coherent sets from sparse and incomplete trajectory data
,” Chaos
25
, 087406
(2015
).25.
A.
Hadjighasem
, D.
Karrasch
, H.
Teramoto
, and G.
Haller
, “A spectral clustering approach to Lagrangian vortex detection
,” preprint arXiv:1506.02258 (2015
).26.
M. O.
Williams
, I. I.
Rypina
, and C. W.
Rowley
, “Identifying finite-time coherent sets from limited quantities of Lagrangian data
,” Chaos
25
, 087408
(2015
).27.
G.
Froyland
, “An analytic framework for identifying finite-time coherent sets in time-dependent dynamical systems
,” Physica D
250
, 1
–19
(2013
).28.
R. R.
Coifman
and S.
Lafon
, “Diffusion maps
,” Appl. Comput. Harmonic Anal.
21
, 5
–30
(2006
).29.
B.
Nadler
, S.
Lafon
, R. R.
Coifman
, and I. G.
Kevrekidis
, “Diffusion maps, spectral clustering and reaction coordinates of dynamical systems
,” Appl. Comput. Harmonic Anal.
21
, 113
–127
(2006
).30.
S.
Lafon
and A. B.
Lee
, “Diffusion maps and coarse-graining: A unified framework for dimensionality reduction, graph partitioning, and data set parameterization
,” IEEE Trans. Pattern Anal. Mach. Intell.
28
, 1393
–1403
(2006
).31.
U.
Von Luxburg
, “A tutorial on spectral clustering
,” Stat. Comput.
17
, 395
–416
(2007
).32.
R.
Lambiotte
, J. C.
Delvenne
, and M.
Barahona
, “Random walks, Markov processes and the multiscale modular organization of complex networks
,” IEEE Trans. Network Sci. Eng.
1
(2
), 76
–90
(2014
).33.
A.
Lasota
and M. C.
Mackey
, Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics
(Springer
, 1994
), Vol. 97.34.
G.
Froyland
and K.
Padberg-Gehle
, “Almost-invariant and finite-time coherent sets: Directionality, duration, and diffusion
,” in Ergodic Theory, Open Dynamics, and Coherent Structures
(Springer
, 2014
), pp. 171
–216
.35.
M.
Hein
, J.-Y.
Audibert
, and U.
von Luxburg
, “From graphs to manifolds – weak and strong pointwise consistency of graph Laplacians
,” in Proceedings of the 18th Annual Conference on Learning Theory, COLT'05
(Springer-Verlag
, Berlin, Heidelberg
, 2005
), pp. 470
–485
.36.
A.
Singer
, “From graph to manifold Laplacian: The convergence rate
,” Appl. Comput. Harmonic Anal.
21
, 128
–134
(2006
).37.
A.
Denner
, O.
Junge
, and D.
Matthes
, “Computing coherent sets using the Fokker–Planck equation
,” preprint arXiv:1512.03761 (2015
).38.
C.
Schütte
, A.
Fischer
, W.
Huisinga
, and P.
Deuflhard
, “A direct approach to conformational dynamics based on hybrid Monte Carlo
,” J. Comput. Phys.
151
, 146
–168
(1999
).39.
P.
Deuflhard
, W.
Huisinga
, A.
Fischer
, and C.
Schütte
, “Identification of almost invariant aggregates in reversible nearly uncoupled Markov chains
,” Linear Algebra Appl.
315
, 39
–59
(2000
).40.
S. E.
Schaeffer
, “Graph clustering
,” Comput. Sci. Rev.
1
, 27
–64
(2007
).41.
J.
Shi
and J.
Malik
, “Normalized cuts and image segmentation
,” IEEE Trans. Pattern Anal. Mach. Intell.
22
, 888
–905
(2000
).42.
A. Y.
Ng
, M. I.
Jordan
, Y.
Weiss
et al, “On spectral clustering: Analysis and an algorithm
,” Advances in Neural Information Processing Systems 14
, edited by T. G.
Dietterich
, S.
Becker
, and Z.
Ghahramani
(MIT Press
, 2002
), pp. 849
–856
, available at http://papers.nips.cc/paper/2092-on-spectral-clustering-analysis-and-an-algorithm.pdf.43.
P.
Deuflhard
and M.
Weber
, “Robust Perron cluster analysis in conformation dynamics
,” Linear Algebra Appl.
398
, 161
–184
(2005
).44.
J. C.
Bezdek
, Pattern Recognition with Fuzzy Objective Function Algorithms
(Springer Science & Business Media
, 2013
).45.
M.
Sarich
, N.
Djurdjevac
, S.
Bruckner
, T. O.
Conrad
, and C.
Schütte
, “Modularity revisited: A novel dynamics-based concept for decomposing complex networks
,” J. Comput. Dyn.
1
, 191
–212
(2014
).46.
S.
Arya
, D. M.
Mount
, N. S.
Netanyahu
, R.
Silverman
, and A. Y.
Wu
, “An optimal algorithm for approximate nearest neighbor searching in fixed dimensions
,” JACM
45
(6
), 891
–923
(1998
).47.
A.
Buluç
and J. R.
Gilbert
, “Parallel sparse matrix-matrix multiplication and indexing: Implementation and experiments
,” SIAM J. Sci. Comput.
34
, C170
–C191
(2012
).48.
W.-Y.
Chen
, Y.
Song
, H.
Bai
, C.-J.
Lin
, and E. Y.
Chang
, “Parallel spectral clustering in distributed systems
,” IEEE Trans. Pattern Anal. Mach. Intell.
33
, 568
–586
(2011
).49.
S.
Klus
, P.
Koltai
, and C.
Schütte
, “On the numerical approximation of the Perron–Frobenius and Koopman operator
,” J. Comput. Dyn.
3
(1
), 51
–79
(2016
).50.
B.
Chen
, B.
Gao
, T.-Y.
Liu
, Y.-F.
Chen
, and W.-Y.
Ma
, “Fast spectral clustering of data using sequential matrix compression
,” Machine Learning: ECML 2006
(Springer
, 2006
), pp. 590
–597
.51.
C.
Fowlkes
, S.
Belongie
, F.
Chung
, and J.
Malik
, “Spectral grouping using the Nystrom method
,” IEEE Trans. Pattern Anal. Mach. Intell.
26
, 214
–225
(2004
).52.
T.-Y.
Liu
, H.-Y.
Yang
, X.
Zheng
, T.
Qin
, and W.-Y.
Ma
, Fast Large-Scale Spectral Clustering by Sequential Shrinkage Optimization
(Springer
, 2007
).53.
C.
Schütte
and M.
Sarich
, Metastability and Markov State Models in Molecular Dynamics
, Courant Lecture Notes in Mathematics (American Mathematical Soc.
, 2013
).54.
G.
Froyland
and P.
Koltai
, “Estimating long-term behavior of periodically driven flows without trajectory integration
,” preprint arXiv:1511.07272 (2015
).55.
V. I.
Arnol'd
, “On the topology of three-dimensional steady flows of an ideal fluid
,” J. Appl. Math. Mech.
30
, 223
–226
(1966
).56.
T.
Dombre
, U.
Frisch
, J. M.
Greene
, M.
Hénon
, A.
Mehr
, and A. M.
Soward
, “Chaotic streamlines in the ABC flows
,” J. Fluid Mech.
167
, 353
(1986
).57.
D.
Blazevski
and G.
Haller
, “Hyperbolic and elliptic transport barriers in three-dimensional unsteady flows
,” Physica D
273
, 46
–62
(2014
).© 2017 Author(s).
2017
Author(s)
You do not currently have access to this content.
