In certain two-dimensional time-dependent flows, the braiding of periodic orbits provides a way to analyze chaos in the system through application of the Thurston-Nielsen classification theorem (TNCT). We expand upon earlier work that introduced the application of the TNCT to braiding of almost-cyclic sets, which are individual components of almost-invariant sets [Stremler et al., “Topological chaos and periodic braiding of almost-cyclic sets,” Phys. Rev. Lett. 106, 114101 (2011)]. In this context, almost-cyclic sets are periodic regions in the flow with high local residence time that act as stirrers or “ghost rods” around which the surrounding fluid appears to be stretched and folded. In the present work, we discuss the bifurcation of the almost-cyclic sets as a system parameter is varied, which results in a sequence of topologically distinct braids. We show that, for Stokes’ flow in a lid-driven cavity, these various braids give good lower bounds on the topological entropy over the respective parameter regimes in which they exist. We make the case that a topological analysis based on spatiotemporal braiding of almost-cyclic sets can be used for analyzing chaos in fluid flows. Hence, we further develop a connection between set-oriented statistical methods and topological methods, which promises to be an important analysis tool in the study of complex systems.
REFERENCES
1.
E.
Ott
, Chaos in Dynamical Systems
, 2nd ed. (Cambridge University Press
, 2002
).2.
S.
Wiggins
, Introduction to Applied Nonlinear Dynamical Systems and Chaos. Texts in Applied Mathematics
(Springer
, Berlin, Heidelberg
, 2003
).3.
H.
Aref
, “Stirring by chaotic advection
,” J. Fluid Mech.
143
, 1
–21
(1984
).4.
H.
Aref
, “The development of chaotic advection
,” Phys. Fluids
14
(4
), 1315
–1325
(2002
).5.
J. M.
Ottino
and S.
Wiggins
, “Introduction: mixing in microfluidics
,” Philos. Trans. R. Soc. London, Ser. A
362
(1818
), 923
–935
(2004
).6.
M.
Stremler
, F.
Haselton
, and H.
Aref
, “Designing for chaos: applications of chaotic advection at the microscale
,” Philos. Trans. R. Soc. London
362
, 1019
–1036
(2004
).7.
8.
A. N.
Kolmogorov
, “New metric invariant of transitive dynamical systems and endomorphisms of Lebesgue spaces
,” Dokl. Phys.
119
(5
), 861
–864
(1958
).9.
Ya. G.
Sinai
, “On the notion of entropy of a dynamical system
,” Dokl. Phys.
124
, 768
–771
(1959
).10.
V.
Rom-Kedar
and S.
Wiggins
, “Transport in two-dimensional maps
,” Arch. Ration. Mech. Anal.
109
, 239
–298
(1990
).11.
S.
Wiggins
, Chaotic Transport in Dynamical Systems, Interdisciplinary Applied Mathematics
(Springer
, Berlin-Heidelberg-New York
, 1992
), Vol. 2
.12.
G.
Haller
, “Distinguished material surfaces and coherent structures in three-dimensional fluid flows
,” Physica D: Nonlinear Phenom.
149
(4
), 248
–277
(2001
).13.
G.
Haller
, “Finding finite-time invariant manifolds in two-dimensional velocity fields
,” Chaos
10
(1
), 99
–108
(2000
).14.
H.
Helmholtz
, “Über die Integrale der hydrodynamischen Gleichungen, welche Wirbelbewegungen entsprechen
,” J. Reine Angew. Math.
55
, 25
–55
(1858
). Translation by P. G. Tait, “On integrals of the hydrodynamical equations which express vortex-motion,” Phil. Mag. 33, 485–512 (1867).15.
R. L.
Ricca
, “Structural complexity and dynamical systems
,” in Lectures on Topological Fluid Mechanics, Lecture Notes in Mathematics
, Vol. 1973
, edited by R. L.
Ricca
, (Springer-Verlag
, Berlin Heidelberg
, 2009
), pp. 167
–186
.16.
P. L.
Boyland
, H.
Aref
, and M. A.
Stremler
“Topological fluid mechanics of stirring
,” J. Fluid Mech.
403
, 277
–304
(2000
).17.
R. L.
Adler
, A. G.
Konheim
, and M. H.
McAndrew
, “Topological entropy
,” Trans. Am. Math. Soc.
114
, 309
–319
(1965
).18.
Y.
Yomdin
, “Volume growth and entropy
,” Isr. J. Math.
57
(3
), 285
–300
(1987
).19.
S. E.
Newhouse
, “Continuity properties of entropy
,” Ann. Math.
129
(2
), 215
–235
(1989
).20.
S. E.
Newhouse
, “Entropy in smooth dynamical systems
,” in Proceedings of the International Congress of Mathematicians, 1991, Vols. 1 and 2, Mathematical Society of Japan, Tokyo (Kyoto, 1990
), pp. 1285
–1294
.21.
S.
Newhouse
and T.
Pignatro
, “On the estimation of topological entropy
,” J. Stat. Phys.
72
(5-6
), 1331
–1351
(1993
).22.
R.
Bowen
, (1978) “Entropy and the fundamental group
,” in The Structure of Attractors in Dynamical Systems
, edited by N. G.
Markley
, J. C.
Martin
, and W.
Perrizo
, Lecture Notes in Mathematics, (Springer
, Berlin
, 1978
), pp. 21
–29
. in Proceedings, June 20–24 (North Dakota State University, 1977), Vol. 668.23.
W.
Thurston
, “On the geometry and dynamics of diffeomorphisms of surfaces
,” Bull. Am. Math. Soc.
19
, 417
(1988
).24.
A. J.
Casson
and S. A.
Bleiler
, Automorphisms of surfaces after Nielsen and Thurston, London Mathematical Society Student Texts
(Cambridge University Press
, Cambridge
, 1988
), Vol. 9
.25.
A.
Fathi
, F.
Laudenbach
, and V.
Poénaru
, Travaux de Thurston sur les surfaces
. Société Mathématique de
, France, Paris
, 1991
. Séminaire Orsay, Reprint of Travaux de Thurston sur les surfaces
, Astérisque No. 66-67, Soc. Math. France, Paris, 1979.26.
A.
Fathi
, F.
Laudenbach
, and V.
Poénaru
, Thurston's Work on Surfaces, Mathematical Notes
(Princeton University Press
, Princeton, New Jersey
, 2012
). Translated by Djun M. Kim and Dan Margalit.27.
L.
Boyland
, “Topological methods in surface dynamics
,” Topol. Appl.
58
(3
), 223
–298
(1994
).28.
M. D.
Finn
, S. M.
Cox
, and H. M.
Byrne
, “Topological chaos in inviscid and viscous mixers
,” J. Fluid Mech.
493
, 345
–361
(2003
).29.
T.
Gouillart
, J.
Thiffeault
, and M.
Finn
, “Topological mixing with ghost rods
,” Phys. Rev. E
73
, 036111
(2006
).30.
M.
Stremler
, and J.
Chen
, “Generating topological chaos in lid-driven cavity flow
,” Phys. Fluids
19
(10
), 103602
(2007
).31.
P. L.
Boyland
, M. A.
Stremler
, and H.
Aref
, “Topological fluid mechanics of point vortex motions
,” Physica D
175
, 69
–95
(2003
).32.
J.
Chen
and M.
Stremler
, “Topological chaos and mixing in a three-dimensional channel flow
,” Phys. Fluids
21
, 021701
(2009
).33.
T.
Hall
, Train: A C++ Program for Computing Train Tracks of Surface Homoemorphism (2001
).34.
M.
Bestvina
and M.
Handel
, “Train-tracks for surface homeomorphisms
,” Topology
34
(1
), 109
–140
(1995
).35.
M.
Handel
, “Gobal shadowing of pseudo-Anosov homeomorphisms
,” Ergod. Theory Dyn. Syst.
5
, 373
–377
(1985
).36.
W.
Magnus
, “Braid group: A survey
,” in Proceedings of the Second International Conference on the Theory of Groups, Lecture notes in Mathematics, (Springer-Verlag
, 1974
), pp. 463
–487
.37.
J. S.
Birman
, Braids, Links and Mapping Class Groups, Annals of Mathematics Studies
(Princeton University Press
, 1975
), Vol. 82
.38.
P. D.
Bangert
, “Braids and knots
,” in Lectures on Topological Fluid Mechanics
, edited by R. L.
Ricca
, Lecture Notes in Mathematics, (Springer-Verlag
, Berlin, Heidelberg
, 2009
), Vol. 1973
, pp. 1
–73
.39.
J.
Thiffeault
, “Measuring topological chaos
,” Phys. Rev. Lett.
94
, 084502
(2005
).40.
J.
Thiffeault
, “Braids of entangled particle trajectories
,” Chaos
20
(1
), 017516
(2010
).41.
M. A.
Stremler
, S. D.
Ross
, P.
Grover
, and P.
Kumar
, “Topological chaos and periodic braiding of almost-cyclic sets
,” Phys. Rev. Lett.
106
, 114101
(2011
).42.
M.
Dellnitz
, and O.
Junge
, “On the approximation of complicated dynamical behavior
,” SIAM J. Numer. Anal.
36
, 491
–515
(1999
).43.
V. V.
Meleshko
and A. M.
Gomilko
, “Infinite systems for a biharmonic problem in a rectangle: further discussion
,” Proc. R. Soc. Lond. Ser. A
460
, 807
–819
(2004
). Our choice of boundary conditions allows for an exact solution with a finite number of terms.44.
W.
Burau
, “Über Zopfgruppen und gleichsinnig verdrilte Verkettungen
,” Abh. Math. Semin. Univ. Hambg.
11
, 171
–178
(1936
).45.
T.
Hall
, and S. O.
Yurttas
, “On the topological entropy of families of braids
,” Topol. Appl.
156
(8
), 1554
–1564
(2009
).46.
S.
Newhouse
and T.
Pignataro
, “On the estimation of topological entropy
,” J. Stat. Phys.
72
, 1331
–1351
(1993
).47.
G.
Froyland
and K.
Padberg
, “Almost-invariant sets and invariant manifolds – connecting probabilistic and geometric descriptions of coherent structures in flows
,” Physica D
238
(16
), 1507
–1523
(2009
).48.
49.
T.
Li
, “Finite approximation for the Fr obenius-Perron operator: A solution to Ulam's conjecture
,” J. Approx. Theory
17
, 177
–186
(1976
).50.
E. M.
Bollt
, A.
Luttman
, S.
Kramer
, and R.
Basnayake
, “Measurable dynamics analysis of transport in the Gulf of Mexico during the oil spill
,” Int. J. Bifurcation Chaos Appl. Sci. Eng.
22
(3
), 1230012
(2012
).51.
E. M.
Bollt
, L.
Billings
, and I. B.
Schwartz
, “A manifold independent approach to understanding transport in stochastic dynamical systems
,” Physica D
173
, 153
–177
(2002
).52.
G.
Froyland
, “Statistically optimal almost-invariant sets
,” Physica D
200
, 205
–219
(2005
).53.
P.
Brémaud
, Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues, Texts in Applied Mathematics
(Springer
, New York
, 1999
).54.
G.
Froyland
and M.
Dellnitz
, “Detecting and locating near-optimal almost-invariant sets and cycles
,” SIAM J. Sci. Comput.
24
, 1839
–1863
(2003
).55.
V.
Rom-Kedar
and S.
Wiggins
, “Transport in two-dimensional maps: Concepts, examples, and a comparison of the theory of Rom-Kedar and Wiggins with the Markov model of MacKay, Meiss, Ott, and Percival
,” Physica D
51
, 248
–266
(1991
).56.
M.
Dellnitz
, O.
Junge
, W. S.
Koon
, F.
Lekien
, M. W.
Lo
, J. E.
Marsden
, K.
Padberg
, R.
Preis
, S. D.
Ross
, and B.
Thiere
, “Transport in dynamical astronomy and multibody problems
,” Int. J. Bifurcation Chaos Appl. Sci. Eng.
15
, 699
–727
(2005
).57.
P. C.
Du Toit
and J. E.
Marsden
, “Horseshoes in hurricanes
,” J. Fixed Point Theory Appl.
7
(2
), 351
–384
(2010
).58.
S. D.
Ross
and P.
Tallapragada
, “Detecting and exploiting chaotic transport in mechanical systems
,” in Applications of Chaos and Nonlinear Dynamics in Science and Engineering
, edited by S.
Banerjee
, L.
Rondoni
, and M.
Mitra
(Springer
, New York
2012
), Vol. 2
.59.
O.
Junge
, J. E.
Marsden
, and I.
Mezic
, “Uncertainty in the dynamics of conservative maps
,” in Proceedings of the 43rd IEEE Conference on Decision and Control
, (2004
), pp. 2225
–2230
.60.
M.
Dellnitz
and I.
Melbourne
, “Generic movement of eigenvalues for equivariant self-adjoint matrices
,” Comput. Appl. Math.
55
, 249
–259
(1994
).61.
See supplementary material at http://dx.doi.org/10.1063/1.4768666 for a video showing transitions between various eigenvectors as τf is changed.
62.
G.
Froyland
, K.
Padberg
, M.
England
, and A.
Treguier
, “Detecting coherent oceanic structures via transfer operators
,” Phys. Rev. Lett.
98
, 224503
(2007
).63.
L.
Billings
and I. B.
Schwartz
, “Identifying almost invariant sets in stochastic dynamical systems
,” Chaos
18
(2
), 023122
(2008
).64.
G.
Froyland
, N.
Santitissadeekorn
, and A.
Monahan
, “Transport in time-dependent dynamical systems: Finite-time coherent sets
,” Chaos
20
, 043116
(2010
).65.
M. K.
Singh
, M. F. M.
Speetjens
, and P. D.
Anderson
, “Eigenmode analysis of scalar transport in distributive mixing
,” Phys. Fluids
21
(9
), 093601
(2009
).66.
W.
Liu
and G.
Haller
, “Strange eigenmodes and decay of variance in the mixing of diffusive tracers
,” Physica D
188
, 1
–39
(2004
).67.
G.
Froyland
, S.
Lloyd
, and N.
Santitissadeekorn
, “Coherent sets for nonautonomous dynamical systems
,” Physica D
239
, 1527
–1541
(2010
).68.
R.
Serban
, W.
Koon
, M.
Lo
, J.
Marsden
, L.
Petzold
, S. D.
Ross
, and R.
Wilson
, “Halo orbit mission correction maneuvers using optimal control
,” Automatica
38
, 571
–583
(2002
).69.
W. S.
Koon
, M. W.
Lo
, J. E.
Marsden
, and S. D.
Ross
, Dynamical Systems, the Three-Body Problem and Space Mission Design
(Marsden Books
, 2008
) ISBN 978-0-615-24095-4.70.
J. E.
Marsden
and S. D.
Ross
, “New methods in celestial mechanics and mission design
,” Bull. Am. Math. Soc.
43
, 43
–73
(2006
).71.
P.
Grover
and S. D.
Ross
, “Designing trajectories in a planet-moon environment using the controlled Keplerian map
,” J. Guid. Control Dyn.
32
, 436
–443
(2009
).72.
F.
Lekien
and S. D.
Ross
, “The computation of finite-time Lyapunov exponents on unstructured meshes and for non-Euclidean manifolds
,” Chaos
20
, 017505
(2010
).73.
G.
Froyland
, O.
Junge
, and P.
Koltai
, “Estimating long term behaviour of flows without trajectory integration: The infinitesimal generator approach,” available at arxiv:1101:4166.© 2012 American Institute of Physics.
2012
American Institute of Physics
You do not currently have access to this content.
