Abstract

Extreme events, such as rogue waves, earthquakes, and stock market crashes, occur spontaneously in many dynamical systems. Because of their usually adverse consequences, quantification, prediction, and mitigation of extreme events are highly desirable. Here, we review several aspects of extreme events in phenomena described by high-dimensional, chaotic dynamical systems. We especially focus on two pressing aspects of the problem: (i) mechanisms underlying the formation of extreme events and (ii) real-time prediction of extreme events. For each aspect, we explore methods relying on models, data, or both. We discuss the strengths and limitations of each approach as well as possible future research directions.

References

References
1.
Kharif
,
C.
, and
Pelinovsky
,
E.
,
2003
, “
Physical Mechanisms of the Rogue Wave Phenomenon
,”
Eur. J. Mech. B/Fluids
,
22
(
6
), pp.
603
634
.10.1016/j.euromechflu.2003.09.002
2.
Dysthe
,
K.
,
Krogstad
,
H. E.
, and
Müller
,
P.
,
2008
, “
Oceanic Rogue Waves
,”
Annu. Rev. Fluid Mech.
,
40
(
1
), pp.
287
310
.10.1146/annurev.fluid.40.111406.102203
3.
Donelan
,
M. A.
, and
Magnusson
,
A.-K.
,
2017
, “
The Making of the Andrea Wave and Other Rogues
,”
Sci. Rep.
,
7
, p.
44124
.10.1038/srep44124
4.
Ropelewski
,
C. F.
, and
Halpert
,
M. S.
,
1987
, “
Global and Regional Scale Precipitation Patterns Associated With the El Niño/Southern Oscillation
,”
Mon. Weather Rev.
,
115
(
8
), pp.
1606
1626
.10.1175/1520-0493(1987)115<1606:GARSPP>2.0.CO;2
5.
Easterling
,
D. R.
,
Evans
,
J.
,
Groisman
,
P. Y.
,
Karl
,
T. R.
,
Kunkel
,
K. E.
, and
Ambenje
,
P.
,
2000
, “
Observed Variability and Trends in Extreme Climate Events: A Brief Review
,”
Bull. Am. Meteorol. Soc.
,
81
(
3
), pp.
417
425
.10.1175/1520-0477(2000)081<0417:OVATIE>2.3.CO;2
6.
Moy
,
C. M.
,
Seltzer
,
G. O.
,
Rodbell
,
D. T.
, and
Anderson
,
D. M.
,
2002
, “
Variability of El Niño/Southern Oscillation Activity at Millennial Timescales During the Holocene Epoch
,”
Nature
,
420
(
6912
), p.
162
.10.1038/nature01194
7.
Dakos
,
V.
,
Scheffer
,
M.
,
van Nes
,
E. H.
,
Brovkin
,
V.
,
Petoukhov
,
V.
, and
Held
,
H.
,
2008
, “
Slowing Down as an Early Warning Signal for Abrupt Climate Change
,”
Proc. Natl. Acad. Sci.
,
105
(
38
), pp.
14308
14312
.10.1073/pnas.0802430105
8.
Geller, R. J.
, 1997, “
Earthquake Prediction: A Critical Review
,”
Geophys. J. Int.
,
131
(3), pp. 425–450.10.1111/j.1365-246X.1997.tb06588.x
9.
Crucitti
,
P.
,
Latora
,
V.
, and
Marchiori
,
M.
,
2004
, “
Model for Cascading Failures in Complex Networks
,”
Phys. Rev. E
,
69
(
4
), p.
045104
.10.1103/PhysRevE.69.045104
10.
Fang
,
X.
,
Misra
,
S.
,
Xue
,
G.
, and
Yang
,
D.
,
2012
, “
Smart Grid—The New and Improved Power Grid: A Survey
,”
IEEE Commun. Surv. Tutorials
,
14
(
4
), pp.
944
980
.10.1109/SURV.2011.101911.00087
11.
Scheffer
,
M.
,
Bascompte
,
J.
,
Brock
,
W. A.
,
Brovkin
,
V.
,
Carpenter
,
S. R.
,
Dakos
,
V.
,
Held
,
H.
,
Van Nes
,
E. H.
,
Rietkerk
,
M.
, and
Sugihara
,
G.
,
2009
, “
Early-Warning Signals for Critical Transitions
,”
Nature
,
461
(
7260
), pp.
53
59
.10.1038/nature08227
12.
Ghil
,
M.
,
Yiou
,
P.
,
Hallegatte
,
S.
,
Malamud
,
B.
,
Naveau
,
P.
,
Soloviev
,
A.
,
Friederichs
,
P.
,
Keilis-Borok
,
V.
,
Kondrashov
,
D.
,
Kossobokov
,
V.
,
Mestre, O.
,
Nicolis, C.
,
Rust, H. W.
,
Shebalin, P.
,
Vrac, M.
,
Witt, A.
, and
Zaliapin, I.
,
2011
, “
Extreme Events: Dynamics, Statistics and Prediction
,”
Nonlinear Processes Geophys.
,
18
(
3
), pp.
295
350
.10.5194/npg-18-295-2011
13.
Dakos
,
V.
,
Carpenter
,
S. R.
,
Brock
,
W. A.
,
Ellison
,
A. M.
,
Guttal
,
V.
,
Ives
,
A. R.
,
Kefi
,
S.
,
Livina
,
V.
,
Seekell
,
D. A.
,
van Nes
,
E. H.
, and
Scheffer, M.
,
2012
, “
Methods for Detecting Early Warnings of Critical Transitions in Time Series Illustrated Using Simulated Ecological Data
,”
PLoS One
,
7
(
7
), p.
e41010
.10.1371/journal.pone.0041010
14.
Ben-Menahem
,
A.
, and
Singh
,
S. J.
,
2012
,
Seismic Waves and Sources
,
Springer Science & Business Media
, New York.
15.
Murphy
,
J. M.
,
Sexton
,
D. M.
,
Barnett
,
D. N.
,
Jones
,
G. S.
,
Webb
,
M. J.
,
Collins
,
M.
, and
Stainforth
,
D. A.
,
2004
, “
Quantification of Modelling Uncertainties in a Large Ensemble of Climate Change Simulations
,”
Nature
,
430
(
7001
), p.
768
.10.1038/nature02771
16.
Scarrott
,
C.
, and
MacDonald
,
A.
,
2012
, “
A Review of Extreme Value Threshold Estimation and Uncertainty Quantification
,”
REVSTAT–Stat. J.
,
10
(
1
), pp.
33
60
.https://www.ine.pt/revstat/pdf/rs120102.pdf
17.
Mohamad
,
M. A.
, and
Sapsis
,
T. P.
,
2015
, “
Probabilistic Description of Extreme Events in Intermittently Unstable Dynamical Systems Excited by Correlated Stochastic Processes
,”
SIAM/ASA J. Uncertainty Quantif.
,
3
(
1
), pp.
709
736
.10.1137/140978235
18.
Doucet
,
A.
,
De Freitas
,
N.
, and
Gordon
,
N.
,
2001
, “
An Introduction to Sequential Monte Carlo Methods
,”
Sequential Monte Carlo Methods in Practice
,
Springer
, New York, pp.
3
14
.
19.
Majda
,
A. J.
, and
Harlim
,
J.
,
2012
,
Filtering Complex Turbulent Systems
,
Cambridge University Press, Cambridge, UK
.
20.
Vanden-Eijnden
,
E.
, and
Weare
,
J.
,
2013
, “
Data Assimilation in the Low Noise Regime With Application to the Kuroshio
,”
Mon. Weather Rev.
,
141
(
6
), pp.
1822
1841
.10.1175/MWR-D-12-00060.1
21.
Altwegg
,
R.
,
Visser
,
V.
,
Bailey
,
L. D.
, and
Erni
,
B.
,
2017
, “
Learning From Single Extreme Events
,”
Philos. Trans. R. Soc. B
,
372
(
1723
), p.
20160141
.10.1098/rstb.2016.0141
22.
Alligood
,
K. T.
,
Sauer
,
T. D.
, and
Yorke
,
J. A.
,
1996
,
Chaos: An Introduction to Dynamical Systems
,
Springer
, New York.
23.
Hirsch
,
M. W.
,
Smale
,
S.
, and
Devaney
,
R. L.
,
2012
,
Differential Equations, Dynamical Systems, and an Introduction to Chaos
,
Academic Press
, New York.
24.
Sornette
,
D.
, and
Ouillon
,
G.
,
2012
, “
Dragon-Kings: Mechanisms, Statistical Methods and Empirical Evidence
,”
Eur. Phys. J. Spec. Top.
,
205
(
1
), pp.
1
26
.10.1140/epjst/e2012-01559-5
25.
Sornette
,
D.
,
2009
, “
Dragon-Kings, Black Swans and the Prediction of Crises
,” preprint
arXiv: 0907.4290.
https://arxiv.org/abs/0907.4290
26.
Turitsyn
,
K.
,
Sulc
,
P.
,
Backhaus
,
S.
, and
Chertkov
,
M.
,
2011
, “
Options for Control of Reactive Power by Distributed Photovoltaic Generators
,”
Proc. IEEE
,
99
(
6
), pp.
1063
1073
.10.1109/JPROC.2011.2116750
27.
Susuki
,
Y.
, and
Mezic
,
I.
,
2012
, “
Nonlinear Koopman Modes and a Precursor to Power System Swing Instabilities
,”
IEEE Trans. Power Syst.
,
27
(
3
), pp.
1182
1191
.10.1109/TPWRS.2012.2183625
28.
Belk
,
J. A.
,
Inam
,
W.
,
Perreault
,
D. J.
, and
Turitsyn
,
K.
,
2016
, “
Stability and Control of Ad Hoc dc Microgrids
,”
IEEE 55th Conference on Decision and Control
(
CDC
), Las Vegas, NV, Dec. 12–14, pp.
3271
3278
.10.1109/CDC.2016.7798761
29.
Hugo
,
L. D.
,
de Cavalcante
,
S.
,
Oriá
,
M.
,
Sornette
,
D.
,
Ott
,
E.
, and
Gauthier
,
D. J.
,
2013
, “
Predictability and Suppression of Extreme Events in a Chaotic System
,”
Phys. Rev. Lett.
,
111
(
19
), p.
198701
.10.1103/PhysRevLett.111.198701
30.
Galuzio
,
P. P.
,
Viana
,
R. L.
, and
Lopes
,
S. R.
,
2014
, “
Control of Extreme Events in the Bubbling Onset of Wave Turbulence
,”
Phys. Rev. E
,
89
(
4
), p.
040901
.10.1103/PhysRevE.89.040901
31.
Chen
,
Y.-Z.
,
Huang
,
Z.-G.
, and
Lai
,
Y.-C.
,
2014
, “
Controlling Extreme Events on Complex Networks
,”
Nat. Sci. Rep.
,
4
, p.
6121
.10.1038/srep06121
32.
Chen
,
Y.-Z.
,
Huang
,
Z.-G.
,
Zhang
,
H.-F.
,
Eisenberg
,
D.
,
Seager
,
T. P.
, and
Lai
,
Y.-C.
,
2015
, “
Extreme Events in Multilayer, Interdependent Complex Networks and Control
,”
Nat. Sci. Rep.
,
5
, p.
17277
.10.1038/srep17277
33.
Bialonski
,
S.
,
Ansmann
,
G.
, and
Kantz
,
H.
,
2015
, “
Data-Driven Prediction and Prevention of Extreme Events in a Spatially Extended Excitable System
,”
Phys. Rev. E
,
92
(
4
), p.
042910
.10.1103/PhysRevE.92.042910
34.
Joo
,
H. K.
,
Mohamad
,
M. A.
, and
Sapsis
,
T. P.
,
2017
, “
Extreme Events and Their Optimal Mitigation in Nonlinear Structural Systems Excited by Stochastic Loads: Application to Ocean Engineering Systems
,”
Ocean Eng.
,
142
, pp.
145
160
.10.1016/j.oceaneng.2017.06.066
35.
Morgan
,
M. G.
,
Henrion
,
M.
, and
Small
,
M.
,
1990
,
Uncertainty: A Guide to Dealing With Uncertainty in Quantitative Risk and Policy Analysis
,
Cambridge University Press
,
Cambridge, UK
.
36.
Wilmott
,
P.
,
2007
,
Paul Wilmott Introduces Quantitative Finance
,
Wiley
, Chichester, UK.
37.
McNeil
,
A. J.
,
Frey
,
R.
, and
Embrechts
,
P.
,
2015
,
Quantitative Risk Management: Concepts, Techniques and Tools
,
Princeton University Press
, Princeton, NJ.
38.
Longin
,
F.
,
2017
,
Extreme Events in Finance: A Handbook of Extreme Value Theory and Its Applications
(Wiley Handbooks in Financial Engineering and Econometrics),
Wiley
, Hoboken, NJ.
39.
de Haan
,
L.
, and
Ferreira
,
A.
,
2007
,
Extreme Value Theory: An Introduction
,
Springer Science & Business Media
, New York.
40.
Fréchet
,
M.
, “
Sur la Loi de Probabilité de L'écart Maximum
,”
Ann. Soc. Polon. Math
,
6
, pp.
93
116
.
41.
Fisher
,
R. A.
, and
Tippett
,
L. H. C.
,
1928
, “
Limiting Forms of the Frequency Distribution of the Largest or Smallest Member of a Sample
,”
Math. Proc. Cambridge Philos. Soc.
,
24
(2), pp.
180
190
.10.1017/S0305004100015681
42.
Gnedenko
,
B.
,
1943
, “
Sur la Distribution Limite du Terme Maximum D'une Serie Aleatoire
,”
Ann. Math.
,
44
(
3
), pp.
423
453
.
43.
Watson
,
G. S.
,
1954
, “
Extreme Values in Samples From m-Dependent Stationary Stochastic Processes
,”
Ann. Math. Stat.
,
25
(
4
), pp.
798
800
.10.1214/aoms/1177728670
44.
Loynes
,
R. M.
,
1965
, “
Extreme Values in Uniformly Mixing Stationary Stochastic Processes
,”
Ann. Math. Stat.
,
36
(
3
), pp.
993
999
.10.1214/aoms/1177700071
45.
Leadbetter
,
M. R.
,
1974
, “
On Extreme Values in Stationary Sequences
,”
Z. Für Wahrscheinlichkeitstheorie Verw. Geb.
,
28
(
4
), pp.
289
303
.10.1007/BF00532947
46.
Leadbetter
,
M. R.
,
1983
, “
Extremes and Local Dependence in Stationary Sequences
,”
Probab. Theory Relat. Fields
,
65
(
2
), pp.
291
306
.10.1007/BF00532484
47.
Hsing
,
T.
,
Hüsler
,
J.
, and
Leadbetter
,
M. R.
,
1988
, “
On the Exceedance Point Process for a Stationary Sequence
,”
Probab. Theory Relat. Fields
,
78
(
1
), pp.
97
112
.10.1007/BF00718038
48.
Leadbetter
,
M. R.
, and
Nandagopalan
,
S.
,
1989
, “
On Exceedance Point Processes for Stationary Sequences Under Mild Oscillation Restrictions
,”
Extreme Value Theory
, Springer, New York, pp.
69
80
.
49.
Chernick
,
M. R.
,
Hsing
,
T.
, and
McCormick
,
W. P.
,
1991
, “
Calculating the Extremal Index for a Class of Stationary Sequences
,”
Adv. Appl. Probab.
,
23
(
4
), pp.
835
850
.10.2307/1427679
50.
Freitas
,
A. C. M.
, and
Freitas
,
J. M.
,
2008
, “
On the Link Between Dependence and Independence in Extreme Value Theory for Dynamical Systems
,”
Stat. Probab. Lett.
,
78
(
9
), pp.
1088
1093
.10.1016/j.spl.2007.11.002
51.
Freitas
,
A. C. M.
,
Freitas
,
J. M.
, and
Todd
,
M.
,
2015
, “
Speed of Convergence for Laws of Rare Events and Escape Rates
,”
Stochastic Processes Appl.
,
125
(
4
), pp.
1653
1687
.10.1016/j.spa.2014.11.011
52.
Lucarini
,
V.
,
Faranda
,
D.
,
Freitas
,
A. C. M.
,
Freitas
,
J. M.
,
Holland
,
M.
,
Kuna
,
T.
,
Nicol
,
M.
,
Todd
,
M.
, and
Vaienti
,
S.
,
2016
,
Extremes and Recurrence in Dynamical Systems
,
Wiley
, Hoboken, NJ.
53.
Cramèr
,
H.
,
1938
, “
Sur un Nouveau Théorème-limite de la Théorie Des Probabilités
,”
Actual. Sci. Ind.
,
736
, pp.
5
23
.
54.
Donsker
,
M. D.
, and
Varadhan
,
S. R. S.
,
1975
, “
Asymptotic Evaluation of Certain Markov Process Expectations for Large Time—I
,”
Commun. Pure Appl. Math.
,
28
(
1
), pp.
1
47
.10.1002/cpa.3160280102
55.
Donsker
,
M. D.
, and
Varadhan
,
S. R. S.
,
1975
, “
Asymptotic Evaluation of Certain Markov Process Expectations for Large Time—II
,”
Commun. Pure Appl. Math.
,
28
(
2
), pp.
279
301
.10.1002/cpa.3160280206
56.
Donsker
,
M. D.
, and
Varadhan
,
S. R. S.
,
1976
, “
Asymptotic Evaluation of Certain Markov Process Expectations for Large Time—III
,”
Commun. Pure Appl. Math.
,
29
(
4
), pp.
389
461
.10.1002/cpa.3160290405
57.
Donsker
,
M. D.
, and
Varadhan
,
S. R. S.
,
1983
, “
Asymptotic Evaluation of Certain Markov Process Expectations for Large Time—IV
,”
Commun. Pure Appl. Math.
,
36
(
2
), pp.
183
212
.10.1002/cpa.3160360204
58.
Varadhan
,
S. R. S.
,
2008
, “
Large Deviations
,”
Ann. Probab.
,
36
(
2
), pp.
397
419
.10.1214/07-AOP348
59.
Touchette
,
H.
,
2009
, “
The Large Deviation Approach to Statistical Mechanics
,”
Phys. Rep.
,
478
(
1–3
), pp.
1
69
.10.1016/j.physrep.2009.05.002
60.
Touchette
,
H.
,
2011
, “
A Basic Introduction to Large Deviations: Theory, Applications, Simulations
,” preprint
arXiv: 1106.4146.
https://arxiv.org/abs/1106.4146
61.
Eyring
,
H.
,
1935
, “
The Activated Complex in Chemical Reactions
,”
J. Chem. Phys.
,
3
(
2
), pp.
107
115
.10.1063/1.1749604
62.
Evans
,
M. G.
, and
Polanyi
,
M.
,
1935
, “
Some Applications of the Transition State Method to the Calculation of Reaction Velocities, Especially in Solution
,”
Trans. Faraday Soc.
,
31
, pp.
875
894
.10.1039/tf9353100875
63.
Laidler
,
K. J.
, and
King
,
M. C.
,
1983
, “
Development of Transition-State Theory
,”
J. Phys. Chem.
,
87
(
15
), pp.
2657
2664
.10.1021/j100238a002
64.
Truhlar
,
D. G.
,
Garrett
,
B. C.
, and
Klippenstein
,
S. J.
,
1996
, “
Current Status of Transition-State Theory
,”
J. Phys. Chem.
,
100
(
31
), pp.
12771
12800
.10.1021/jp953748q
65.
Vanden-Eijnden
,
E.
,
2006
, “
Transition Path Theory
,”
Computer Simulations in Condensed Matter Systems: From Materials to Chemical Biology
, Vol.
1
, Springer, Berlin, pp.
453
493
.
66.
Metzner
,
P.
,
Schütte
,
C.
, and
Vanden-Eijnden
,
E.
,
2006
, “
Illustration of Transition Path Theory on a Collection of Simple Examples
,”
J. Chem. Phys.
,
125
(
8
), p.
084110
.10.1063/1.2335447
67.
Weinan
,
E.
, and
Vanden-Eijnden
,
E.
,
2010
, “
Transition-Path Theory and Path-Finding Algorithms for the Study of Rare Events
,”
Annu. Rev. Phys. Chem.
,
61
(
1
), pp.
391
420
.10.1146/annurev.physchem.040808.090412
68.
van der Pol
,
B.
,
1926
, “
LXXXVIII—on ‘Relaxation-Oscillations'
,”
London, Edinburgh, Dublin Philos. Mag. J. Sci.
,
2
(
11
), pp.
978
992
.10.1080/14786442608564127
69.
van der Pol
,
B.
,
1934
, “
The Nonlinear Theory of Electric Oscillations
,”
Proc. Inst. Radio Eng.
,
22
(
9
), pp.
1051
1086
.10.1109/JRPROC.1934.226781
70.
Benoît
,
E.
,
1983
, “
Systèmes Lents-rapides Dans ℝ3 et Leurs Canards
,”
Astérisque
,
109–110
, pp.
159
191
.
71.
Field
,
R. J.
, and
Noyes
,
R. M.
,
1974
, “
Oscillations in Chemical Systems—IV: Limit Cycle Behavior in a Model of a Real Chemical Reaction
,”
J. Chem. Phys.
,
60
(
5
), pp.
1877
1884
.10.1063/1.1681288
72.
Gillespie
,
D. T.
,
1977
, “
Exact Stochastic Simulation of Coupled Chemical Reactions
,”
J. Phys. Chem.
,
81
(
25
), pp.
2340
2361
.10.1021/j100540a008
73.
Connors
,
K. A.
,
1990
,
Chemical Kinetics: The Study of Reaction Rates in Solution
,
Wiley
, Hoboken, NJ.
74.
Koper
,
M. T. M.
, and
Gaspard
,
P.
,
1991
, “
Mixed-Mode and Chaotic Oscillations in a Simple Model of an Electrochemical Oscillator
,”
J. Phys. Chem.
,
95
(
13
), pp.
4945
4947
.10.1021/j100166a009
75.
Arneodo
,
A.
, and
Elezgaray
,
J.
,
1995
, “
Modeling Front Pattern Formation and Intermittent Bursting Phenomena in the Couette Flow Reactor
,”
Chemical Waves and Patterns
,
R.
Kapral
and
K.
Showalter
, eds.,
Springer
, Dordrecht,
The Netherlands
, pp.
517
570
.
76.
Ermentrout
,
G. B.
, and
Kopell
,
N.
,
1986
, “
Parabolic Bursting in an Excitable System Coupled With a Slow Oscillation
,”
SIAM J. Appl. Math.
,
46
(
2
), pp.
233
253
.10.1137/0146017
77.
Rinzel
,
J.
,
1987
, “
A Formal Classification of Bursting Mechanisms in Excitable Systems
,”
Mathematical Topics in Population Biology, Morphogenesis and Neurosciences
,
Springer
, New York, pp.
267
281
.
78.
Izhikevich
,
E. M.
,
2000
, “
Neural Excitability, Spiking and Bursting
,”
Int. J. Bifurcation Chaos
,
10
(
06
), pp.
1171
1266
.10.1142/S0218127400000840
79.
Guckenheimer
,
J.
, and
Oliva
,
R. A.
,
2002
, “
Chaos in the Hodgkin–Huxley Model
,”
SIAM J. Appl. Dyn. Syst.
,
1
(
1
), pp.
105
114
.10.1137/S1111111101394040
80.
Ansmann
,
G.
,
Karnatak
,
R.
,
Lehnertz
,
K.
, and
Feudel
,
U.
,
2013
, “
Extreme Events in Excitable Systems and Mechanisms of Their Generation
,”
Phys. Rev. E
,
88
(
5
), p.
052911
.10.1103/PhysRevE.88.052911
81.
Karnatak
,
R.
,
Ansmann
,
G.
,
Feudel
,
U.
, and
Lehnertz
,
K.
,
2014
, “
Route to Extreme Events in Excitable Systems
,”
Phys. Rev. E
,
90
(
2
), p.
022917
.10.1103/PhysRevE.90.022917
82.
Saha
,
A.
, and
Feudel
,
U.
,
2017
, “
Extreme Events in FitzHugh-Nagumo Oscillators Coupled With Two Time Delays
,”
Phys. Rev. E
,
95
(
6
), p.
062219
.10.1103/PhysRevE.95.062219
83.
Latif
,
M.
, and
Keenlyside
,
N. S.
,
2009
, “
El Niño/Southern Oscillation Response to Global Warming
,”
Proc. Natl. Acad. Sci.
,
106
(
49
), pp.
20578
20583
.10.1073/pnas.0710860105
84.
Dijkstra
,
H. A.
,
2013
,
Nonlinear Climate Dynamics
,
Cambridge University Press
,
Cambridge, UK
.
85.
Roberts
,
A.
,
Guckenheimer
,
J.
,
Widiasih
,
E.
,
Timmermann
,
A.
, and
Jones
,
C. K. R. T.
,
2016
, “
Mixed-Mode Oscillations of El Niño–Southern Oscillation
,”
J. Atmos. Sci.
,
73
(
4
), pp.
1755
1766
.10.1175/JAS-D-15-0191.1
86.
Haller
,
G.
, and
Sapsis
,
T.
,
2010
, “
Localized Instability and Attraction Along Invariant Manifolds
,”
SIAM J. Appl. Dyn. Syst.
,
9
(
2
), pp.
611
633
.10.1137/08074324X
87.
Sapsis
,
T.
, and
Haller
,
G.
,
2008
, “
Instabilities in the Dynamics of Neutrally Buoyant Particles
,”
Phys. Fluids
,
20
(
1
), p.
017102
.10.1063/1.2830328
88.
Wiggins
,
S.
,
1994
, “
Normally Hyperbolic Invariant Manifolds in Dynamical Systems
,”
Applied Mathematical Sciences
, Vol.
105
,
Springer
, New York.
89.
Jones
,
C. K. R. T.
,
1995
, “
Geometric Singular Perturbation Theory
,”
Dynamical Systems
,
Springer
, New York, pp.
44
118
.
90.
Desroches
,
M.
,
Guckenheimer
,
J.
,
Krauskopf
,
B.
,
Kuehn
,
C.
,
Osinga
,
H. M.
, and
Wechselberger
,
M.
,
2012
, “
Mixed-Mode Oscillations With Multiple Time Scales
,”
SIAM Rev.
,
54
(
2
), pp.
211
288
.10.1137/100791233
91.
Fenichel
,
N.
,
1979
, “
Geometric Singular Perturbation Theory for Ordinary Differential Equations
,”
J. Differ. Equations
,
31
(
1
), pp.
53
98
.10.1016/0022-0396(79)90152-9
92.
Guckenheimer
,
J.
,
2008
, “
Singular Hopf Bifurcation in Systems With Two Slow Variables
,”
SIAM J. Appl. Dyn. Syst.
,
7
(
4
), pp.
1355
1377
.10.1137/080718528
93.
Guckenheimer
,
J.
, and
Vladimirsky
,
A.
,
2004
, “
A Fast Method for Approximating Invariant Manifolds
,”
SIAM J. Appl. Dyn. Syst.
,
3
(
3
), pp.
232
260
.10.1137/030600179
94.
Krauskopf
,
B.
,
Osinga
,
H. M.
,
Doedel
,
E. J.
,
Henderson
,
M. E.
,
Guckenheimer
,
J.
,
Vladimirsky
,
A.
,
Dellnitz
,
M.
, and
Junge
,
O.
,
2005
, “
A Survey of Methods for Computing (Un)stable Manifolds of Vector Fields
,”
Int. J. Bifurcation Chaos
,
15
(
3
), pp.
763
791
.10.1142/9789812774569_0004
95.
Lebiedz
,
D.
,
Siehr
,
J.
, and
Unger
,
J.
,
2011
, “
A Variational Principle for Computing Slow Invariant Manifolds in Dissipative Dynamical Systems
,”
SIAM J. Sci. Comput.
,
33
(
2
), pp.
703
720
.10.1137/100790318
96.
Castelli
,
R.
,
Lessard
,
J.-P.
, and
James
,
J. D. M.
,
2015
, “
Parameterization of Invariant Manifolds for Periodic Orbits—I: Efficient Numerics Via the Floquet Normal Form
,”
SIAM J. Appl. Dyn. Syst.
,
14
(
1
), pp.
132
167
.10.1137/140960207
97.
Babaee
,
H.
,
Farazmand
,
M.
,
Haller
,
G.
, and
Sapsis
,
T. P.
,
2017
, “
Reduced-Order Description of Transient Instabilities and Computation of Finite-Time Lyapunov Exponents
,”
Chaos
,
27
(
6
), p.
063103
.10.1063/1.4984627
98.
Shilnikov
,
L. P.
,
1965
, “
A Case of the Existence of a Denumerable Set of Periodic Motions
,”
Sov. Math. Dokl.
,
6
, pp.
163
166
.
99.
Gaspard
,
P.
, and
Nicolis
,
G.
,
1983
, “
What Can We Learn From Homoclinic Orbits in Chaotic Dynamics?
,”
J. Stat. Phys.
,
31
(
3
), pp.
499
518
.10.1007/BF01019496
100.
Cvitanović
,
P.
, and
Eckhardt
,
B.
,
1989
, “
Periodic-Orbit Quantization of Chaotic Systems
,”
Phys. Rev. Lett.
,
63
(
8
), p.
823
.10.1103/PhysRevLett.63.823
101.
Cvitanović
,
P.
,
1991
, “
Periodic Orbits as the Skeleton of Classical and Quantum Chaos
,”
Phys. D: Nonlinear Phenom.
,
51
(
1–3
), pp.
138
151
.10.1016/0167-2789(91)90227-Z
102.
Rössler
,
O.
,
1976
, “
An Equation for Continuous Chaos
,”
Phys. Lett. A
,
57
(
5
), pp.
397
398
.10.1016/0375-9601(76)90101-8
103.
Letellier
,
C.
,
Dutertre
,
P.
, and
Maheu
,
B.
,
1995
, “
Unstable Periodic Orbits and Templates of the Rössler System: Toward a Systematic Topological Characterization
,”
Chaos
,
5
(
1
), pp.
271
282
.10.1063/1.166076
104.
Meacham
,
S. P.
,
2000
, “
Low-Frequency Variability in the Wind-Driven Circulation
,”
J. Phys. Oceanogr.
,
30
(
2
), pp.
269
293
.10.1175/1520-0485(2000)030<0269:LFVITW>2.0.CO;2
105.
Timmermann
,
A.
,
Jin
,
F.-F.
, and
Abshagen
,
J.
,
2003
, “
A Nonlinear Theory for El Niño Bursting
,”
J. Atmos. Sci.
,
60
(
1
), pp.
152
165
.10.1175/1520-0469(2003)060<0152:ANTFEN>2.0.CO;2
106.
Ermentrout
,
B.
,
1998
, “
Neural Networks as Spatio-Temporal Pattern-Forming Systems
,”
Rep. Prog. Phys.
,
61
(
4
), p.
353
.10.1088/0034-4885/61/4/002
107.
Izhikevich
,
E. M.
,
2003
, “
Simple Model of Spiking Neurons
,”
IEEE Trans. Neural Networks
,
14
(
6
), pp.
1569
1572
.10.1109/TNN.2003.820440
108.
Elezgaray
,
J.
, and
Arneodo
,
A.
,
1992
, “
Crisis-Induced Intermittent Bursting in Reaction-Diffusion Chemical Systems
,”
Phys. Rev. Lett.
,
68
(
5
), p.
714
.10.1103/PhysRevLett.68.714
109.
Farazmand
,
M.
, and
Sapsis
,
T. P.
,
2016
, “
Dynamical Indicators for the Prediction of Bursting Phenomena in High-Dimensional Systems
,”
Phys. Rev. E
,
94
(
3–1
), p.
032212
.10.1103/PhysRevE.94.032212
110.
Coller
,
B.
,
Holmes
,
P.
, and
Lumley
,
J.
,
1994
, “
Interaction of Adjacent Bursts in the Wall Region
,”
Phys. Fluids
,
6
(
2
), pp.
954
961
.10.1063/1.868425
111.
Jones
,
C.
, and
Kopell
,
N.
,
1994
, “
Tracking Invariant Manifolds With Differential Forms in Singularly Perturbed Systems
,”
J. Differ. Equations
,
108
(
1
), pp.
64
88
.10.1006/jdeq.1994.1025
112.
Han
,
S. K.
,
Kurrer
,
C.
, and
Kuramoto
,
Y.
,
1995
, “
Dephasing and Bursting in Coupled Neural Oscillators
,”
Phys. Rev. Lett.
,
75
(
17
), p.
3190
.10.1103/PhysRevLett.75.3190
113.
Haller
,
G.
, and
Wiggins
,
S.
,
1995
, “
Multi-Pulse Jumping Orbits and Homoclinic Trees in a Modal Truncation of the Damped-Forced Nonlinear Schrödinger Equation
,”
Phys. D: Nonlinear Phenom.
,
85
(
3
), pp.
311
347
.10.1016/0167-2789(95)00120-S
114.
Coller
,
B.
, and
Holmes
,
P.
,
1997
, “
Suppression of Bursting
,”
Automatica
,
33
(
1
), pp.
1
11
.10.1016/S0005-1098(96)00137-9
115.
Kawahara
,
G.
, and
Kida
,
S.
,
2001
, “
Periodic Motion Embedded in Plane Couette Turbulence: Regeneration Cycle and Burst
,”
J. Fluid Mech.
,
449
, pp.
291
300
.10.1017/S0022112001006243
116.
Farazmand
,
M.
,
2016
, “
An Adjoint-Based Approach for Finding Invariant Solutions of Navier–Stokes Equations
,”
J. Fluid Mech.
,
795
, pp.
278
312
.10.1017/jfm.2016.203
117.
Horsthemke
,
W.
,
1984
, “
Noise Induced Transitions
,”
Non-Equilibrium Dynamics in Chemical Systems
,
Springer
, New York, pp.
150
160
.
118.
Van den Broeck
,
C.
,
Parrondo
,
J. M. R.
, and
Toral
,
R.
,
1994
, “
Noise-Induced Nonequilibrium Phase Transition
,”
Phys. Rev. Lett.
,
73
(
25
), p.
3395
.10.1103/PhysRevLett.73.3395
119.
Neiman
,
A. B.
, and
Russell
,
D. F.
,
2002
, “
Synchronization of Noise-Induced Bursts in Noncoupled Sensory Neurons
,”
Phys. Rev. Lett.
,
88
(
13
), p.
138103
.10.1103/PhysRevLett.88.138103
120.
Moore
,
R. O.
,
Biondini
,
G.
, and
Kath
,
W. L.
,
2008
, “
A Method to Compute Statistics of Large, Noise-Induced Perturbations of Nonlinear Schrödinger Solitons
,”
SIAM Rev.
,
50
(
3
), pp.
523
549
.10.1137/080722977
121.
Forgoston
,
E.
, and
Moore
,
R. O.
,
2017
, “
A Primer on Noise-Induced Transitions in Applied Dynamical Systems
,”
SIAM Rev.
,
60
(4), pp. 969–1009.10.1137/17M1142028
122.
Wigner
,
E.
,
1938
, “
The Transition State Method
,”
Trans. Faraday Soc.
,
34
, pp.
29
41
.10.1039/tf9383400029
123.
Horiuti
,
J.
,
1938
, “
On the Statistical Mechanical Treatment of the Absolute Rate of Chemical Reaction
,”
Bull. Chem. Soc. Jpn.
,
13
(
1
), pp.
210
216
.10.1246/bcsj.13.210
124.
Yamamoto
,
T.
,
1960
, “
Quantum Statistical Mechanical Theory of the Rate of Exchange Chemical Reactions in the Gas Phase
,”
J. Chem. Phys.
,
33
(
1
), pp.
281
289
.10.1063/1.1731099
125.
Chandler
,
D.
,
1978
, “
Statistical Mechanics of Isomerization Dynamics in Liquids and the Transition State Approximation
,”
J. Chem. Phys.
,
68
(
6
), pp.
2959
2970
.10.1063/1.436049
126.
Pratt
,
L. R.
,
1986
, “
A Statistical Method for Identifying Transition States in High Dimensional Problems
,”
J. Chem. Phys.
,
85
(
9
), pp.
5045
5048
.10.1063/1.451695
127.
Weinan
,
E.
, and
Vanden-Eijnden
,
E.
,
2006
, “
Towards a Theory of Transition Paths
,”
J. Stat. Phys.
,
123
(
3
), p.
503
.10.1007/s10955-005-9003-9
128.
Gonzalez
,
C.
, and
Schlegel
,
H. B.
,
1989
, “
An Improved Algorithm for Reaction Path Following
,”
J. Chem. Phys.
,
90
(
4
), pp.
2154
2161
.10.1063/1.456010
129.
Bolhuis
,
P. G.
,
Chandler
,
D.
,
Dellago
,
C.
, and
Geissler
,
P. L.
,
2002
, “
Transition Path Sampling: Throwing Ropes Over Rough Mountain Passes, in the Dark
,”
Annu. Rev. Phys. Chem.
,
53
(
1
), pp.
291
318
.10.1146/annurev.physchem.53.082301.113146
130.
Dellago
,
C.
,
Bolhuis
,
P.
, and
Geissler
,
P. L.
,
2002
, “
Transition Path Sampling
,”
Adv. Chem. Phys.
,
123
(
1
), pp. 1–78.
131.
Weinan
,
E.
,
Ren
,
W.
, and
Vanden-Eijnden
,
E.
,
2002
, “
String Method for the Study of Rare Events
,”
Phys. Rev. B
,
66
(
5
), p.
052301
.10.1103/PhysRevB.66.052301
132.
Maragliano
,
L.
,
Fischer
,
A.
,
Vanden-Eijnden
,
E.
, and
Ciccotti
,
G.
,
2006
, “
String Method in Collective Variables: Minimum Free Energy Paths and Isocommittor Surfaces
,”
J. Chem. Phys.
,
125
(
2
), p.
024106
.10.1063/1.2212942
133.
Weinan
,
E.
,
Ren
,
W.
, and
Vanden-Eijnden
,
E.
,
2007
, “
Simplified and Improved String Method for Computing the Minimum Energy Paths in Barrier-Crossing Events
,”
J. Chem. Phys.
,
126
(
16
), p.
164103
.10.1063/1.2720838
134.
Pan
,
A. C.
,
Sezer
,
D.
, and
Roux
,
B.
,
2008
, “
Finding Transition Pathways Using the String Method With Swarms of Trajectories
,”
J. Phys. Chem. B
,
112
(
11
), pp.
3432
3440
.10.1021/jp0777059
135.
Farazmand
,
M.
, and
Sapsis
,
T. P.
,
2017
, “
A Variational Approach to Probing Extreme Events in Turbulent Dynamical Systems
,”
Sci. Adv.
,
3
(
9
), p.
e1701533
.10.1126/sciadv.1701533
136.
Ruelle
,
D.
,
1989
,
Chaotic Evolution and Strange Attractors
, Vol.
1
,
Cambridge University Press
,
Cambridge, UK
.
137.
Constantin
,
P.
,
Foias
,
C.
,
Nicolaenko
,
B.
, and
Temam
,
R.
,
1989
,
Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations
(Applied Mathematical Sciences, Vol.
70
), Springer, New York.
138.
Dellnitz
,
M.
, and
Junge
,
O.
,
2004
,
On the Approximation of Complicated Dynamical Behavior
,
Springer
,
New York
, pp.
400
424
.
139.
Chen
,
K. K.
,
Tu
,
J. H.
, and
Rowley
,
C. W.
,
2012
, “
Variants of Dynamic Mode Decomposition: Boundary Condition, Koopman, and Fourier Analyses
,”
J. Nonlinear Sci.
,
22
(
6
), pp.
887
915
.10.1007/s00332-012-9130-9
140.
Hinze
,
M.
,
Pinnau
,
R.
,
Ulbrich
,
M.
, and
Ulbrich
,
S.
,
2008
, “
Optimization With PDE Constraints
,”
Mathematical Modeling: Theory and Applications
, Vol.
23
,
Springer Science & Business Media
, Dordrecht, The Netherlands.
141.
Herzog
,
R.
, and
Kunisch
,
K.
,
2010
, “
Algorithms for PDE-Constrained Optimization
,”
GAMM-Mitteilungen
,
33
(
2
), pp.
163
176
.10.1002/gamm.201010013
142.
Farazmand
,
M.
,
Kevlahan
,
N. K.-R.
, and
Protas
,
B.
,
2011
, “
Controlling the Dual Cascade of Two-Dimensional Turbulence
,”
J. Fluid Mech.
,
668
, pp.
202
222
.10.1017/S0022112010004635
143.
Bertsekas
,
D. P.
,
2014
,
Constrained Optimization and Lagrange Multiplier Methods
,
Academic Press
, New York.
144.
Obukhov
,
A. M.
,
1983
, “
Kolmogorov Flow and Laboratory Simulation of It
,”
Russ. Math. Surv.
,
38
(
4
), pp.
113
126
.10.1070/RM1983v038n04ABEH004207
145.
Marchioro
,
C.
,
1986
, “
An Example of Absence of Turbulence for Any Reynolds Number
,”
Commun. Math. Phys.
,
105
(
1
), pp.
99
106
.10.1007/BF01212343
146.
Platt
,
N.
,
Sirovich
,
L.
, and
Fitzmaurice
,
N.
,
1991
, “
An Investigation of Chaotic Kolmogorov Flows
,”
Phys. Fluids A
,
3
(
4
), pp.
681
696
.10.1063/1.858074
147.
Foias
,
C.
,
Manley
,
O.
,
Rosa
,
R.
, and
Temam
,
R.
,
2001
,
Navier–Stokes Equations and Turbulence
,
Cambridge University Press
,
Cambridge, UK
.
148.
Chandler
,
G. J.
, and
Kerswell
,
R. R.
,
2013
, “
Invariant Recurrent Solutions Embedded in a Turbulent Two-Dimensional Kolmogorov Flow
,”
J. Fluid Mech.
,
722
, pp.
554
595
.10.1017/jfm.2013.122
149.
Batchaev
,
A.
, and
Dovzhenko
,
V.
,
1983
, “
Laboratory Simulation of the Stability Loss of Periodic Zonal Flows
,”
Akademiia Nauk SSSR Doklady
, Vol.
273
, Doklady, Moscow, Russia, pp.
582
584
.
150.
Burgess
,
J. M.
,
Bizon
,
C.
,
McCormick
,
W.
,
Swift
,
J.
, and
Swinney
,
H. L.
,
1999
, “
Instability of the Kolmogorov Flow in a Soap Film
,”
Phys. Rev. E
,
60
(
1
), p.
715
.10.1103/PhysRevE.60.715
151.
Ouellette
,
N. T.
, and
Gollub
,
J. P.
,
2008
, “
Dynamic Topology in Spatiotemporal Chaos
,”
Phys. Fluids
,
20
(
6
), p.
064104
.10.1063/1.2948849
152.
Suri
,
B.
,
Tithof
,
J.
,
Grigoriev
,
R. O.
, and
Schatz
,
M. F.
,
2017
, “
Forecasting Fluid Flows Using the Geometry of Turbulence
,”
Phys. Rev. Lett.
,
118
(
11
), p.
114501
.10.1103/PhysRevLett.118.114501
153.
Majda
,
A.
,
Abramov
,
R. V.
, and
Grote
,
M. J.
,
2005
,
Information Theory and Stochastics for Multiscale Nonlinear Systems
, Vol.
25
,
American Mathematical Society
, Providence, RI.
154.
Kraichnan
,
R. H.
,
1971
, “
Inertial-Range Transfer in Two- and Three-Dimensional Turbulence
,”
J. Fluid Mech.
,
47
(
3
), pp.
525
535
.10.1017/S0022112071001216
155.
Moffatt
,
H. K.
,
2014
, “
Note on the Triad Interactions of Homogeneous Turbulence
,”
J. Fluid Mech.
,
741
, p. R3.10.1017/jfm.2013.637
156.
Stoker
,
J. J.
,
1958
,
Water Waves: The Mathematical Theory With Applications
,
Wiley
, New York.
157.
Nieto Borge
,
J. C.
,
RodrÍguez
,
G. R.
,
Hessner
,
K.
, and
González
,
P. I.
,
2004
, “
Inversion of Marine Radar Images for Surface Wave Analysis
,”
J. Atmos. Oceanic Technol.
,
21
(
8
), pp.
1291
1300
.10.1175/1520-0426(2004)021<1291:IOMRIF>2.0.CO;2
158.
Fu
,
T. C.
,
Fullerton
,
A. M.
,
Hackett
,
E. E.
, and
Merrill
,
C.
,
2011
, “
Shipboard Measurments of Ocean Waves
,”
ASME
Paper No. OMAE2011-49894.10.1115/OMAE2011-49894
159.
Story
,
W. R.
,
Fu
,
T. C.
, and
Hackett
,
E. E.
,
2011
, “
Radar Measurement of Ocean Waves
,”
ASME
Paper No. OMAE2011-49895.10.1115/OMAE2011-49895
160.
Nieto Borge
,
J. C.
,
Reichert
,
K.
, and
Hessner
,
K.
,
2013
, “
Detection of Spatio-Temporal Wave Grouping Properties by Using Temporal Sequences of X-Band Radar Images of the Sea Surface
,”
Ocean Modell.
,
61
, pp.
21
37
.10.1016/j.ocemod.2012.10.004
161.
Benney
,
D. J.
, and
Newell
,
A. C.
,
1967
, “
The Propagation of Nonlinear Wave Envelopes
,”
J. Math. Phys.
,
46
(
1–4
), pp.
133
139
.10.1002/sapm1967461133
162.
Zakharov
,
V. E.
,
1968
, “
Stability of Periodic Waves of Finite Amplitude on the Surface of a Deep Fluid
,”
J. Appl. Mech. Tech. Phys.
,
9
(
2
), pp.
190
194
.10.1007/BF00913182
163.
Hasimoto
,
H.
, and
Ono
,
H.
,
1972
, “
Nonlinear Modulation of Gravity Waves
,”
J. Phys. Soc. Jpn.
,
33
(
3
), pp.
805
811
.10.1143/JPSJ.33.805
164.
Yuen
,
H. C.
, and
Lake
,
B. M.
,
1975
, “
Nonlinear Deep Water Waves: Theory and Experiment
,”
Phys. Fluids
,
18
(
8
), pp.
956
960
.10.1063/1.861268
165.
Dysthe
,
K. B.
,
1979
, “
Note on a Modification to the Nonlinear Schrödinger Equation for Application to Deep Water Waves
,”
Proc. R. Soc. A
,
369
(
1736
), pp.
105
114
.10.1098/rspa.1979.0154
166.
Trulsen
,
K.
,
Kliakhandler
,
I.
,
Dysthe
,
K. B.
, and
Velarde
,
M. G.
,
2000
, “
On Weakly Nonlinear Modulation of Waves on Deep Water
,”
Phys. Fluids
,
12
(
10
), pp.
2432
2437
.10.1063/1.1287856
167.
Ma
,
Y.-C.
,
1979
, “
The Perturbed Plane-Wave Solutions of the Cubic Schrödinger Equation
,”
Stud. Appl. Math.
,
60
(
1
), pp.
43
58
.10.1002/sapm197960143
168.
Akhmediev
,
N. N.
, and
Korneev
,
V. I.
,
1986
, “
Modulation Instability and Periodic Solutions of the Nonlinear Schrödinger Equation
,”
Theor. Math. Phys.
,
69
(
2
), pp.
1089
1093
.10.1007/BF01037866
169.
Peregrine
,
D.
,
1983
, “
Water Waves, Nonlinear Schrödinger Equations and Their Solutions
,”
J. Aust. Math. Soc. Ser. B. Appl. Math.
,
25
(
1
), pp.
16
43
.10.1017/S0334270000003891
170.
Akhmediev
,
N.
,
Ankiewicz
,
A.
, and
Soto-Crespo
,
J. M.
,
2009
, “
Rogue Waves and Rational Solutions of the Nonlinear Schrödinger Equation
,”
Phys. Rev. E
,
80
, p.
026601
.10.1103/PhysRevE.80.026601
171.
Baronio
,
F.
,
Chen
,
S.
,
Grelu
,
P.
,
Wabnitz
,
S.
, and
Conforti
,
M.
,
2015
, “
Baseband Modulation Instability as the Origin of Rogue Waves
,”
Phys. Rev. A
,
91
, p.
033804
.10.1103/PhysRevA.91.033804
172.
Zhao
,
L.-C.
, and
Ling
,
L.
,
2016
, “
Quantitative Relations Between Modulational Instability and Several Well-Known Nonlinear Excitations
,”
J. Opt. Soc. Am. B
,
33
(
5
), pp.
850
856
.10.1364/JOSAB.33.000850
173.
Wen
,
X.-Y.
,
Yan
,
Z.
, and
Yang
,
Y.
,
2016
, “
Dynamics of Higher-Order Rational Solitons for the Nonlocal Nonlinear Schrödinger Equation With the Self-Induced Parity-Time-Symmetric Potential
,”
Chaos
,
26
(
6
), p.
063123
.10.1063/1.4954767
174.
Chen
,
S.
,
Baronio
,
F.
,
Soto-Crespo
,
J. M.
,
Grelu
,
P.
, and
Mihalache
,
D.
,
2017
, “
Versatile Rogue Waves in Scalar, Vector, and Multidimensional Nonlinear Systems
,”
J. Phys. A: Math. Theor.
,
50
(
46
), p.
463001
.10.1088/1751-8121/aa8f00
175.
Bertola
,
M.
, and
Tovbis
,
A.
,
2013
, “
Universality for the Focusing Nonlinear Schrödinger Equation at the Gradient Catastrophe Point: Rational Breathers and Poles of the Tritronquée Solution to Painlevé I
,”
Commun. Pure Appl. Math.
,
66
(
5
), pp.
678
752
.10.1002/cpa.21445
176.
Tikan
,
A.
,
Billet
,
C.
,
El
,
G.
,
Tovbis
,
A.
,
Bertola
,
M.
,
Sylvestre
,
T.
,
Gustave
,
F.
,
Randoux
,
S.
,
Genty
,
G.
,
Suret
,
P.
, and
Dudley
,
J. M.
,
2017
, “
Universality of the Peregrine Soliton in the Focusing Dynamics of the Cubic Nonlinear Schrödinger Equation
,”
Phys. Rev. Lett.
,
119
(
3
), p.
033901
.10.1103/PhysRevLett.119.033901
177.
Chabchoub
,
A.
,
Hoffmann
,
N. P.
, and
Akhmediev
,
N.
,
2011
, “
Rogue Wave Observation in a Water Wave Tank
,”
Phys. Rev. Lett.
,
106
(
20
), p.
204502
.10.1103/PhysRevLett.106.204502
178.
Chabchoub
,
A.
,
Hoffmann
,
N.
,
Onorato
,
M.
, and
Akhmediev
,
N.
,
2012
, “
Super Rogue Waves: Observation of a Higher-Order Breather in Water Waves
,”
Phys. Rev. X
,
2
(
1
), p.
011015
.10.1103/PhysRevX.2.011015
179.
Chabchoub
,
A.
,
Akhmediev
,
N.
, and
Hoffmann
,
N.
,
2012
, “
Experimental Study of Spatiotemporally Localized Surface Gravity Water Waves
,”
Phys. Rev. E
,
86
(
1
), p.
016311
.10.1103/PhysRevE.86.016311
180.
Chabchoub
,
A.
,
Kimmoun
,
O.
,
Branger
,
H.
,
Hoffmann
,
N.
,
Proment
,
D.
,
Onorato
,
M.
, and
Akhmediev
,
N.
,
2013
, “
Experimental Observation of Dark Solitons on the Surface of Water
,”
Phys. Rev. Lett.
,
110
(
12
), p.
124101
.10.1103/PhysRevLett.110.124101
181.
Shemer
,
L.
, and
Alperovich
,
L.
,
2013
, “
Peregrine Breather Revisited
,”
Phys. Fluids
,
25
(
5
), p.
051701
.10.1063/1.4807055
182.
Kimmoun
,
O.
,
Hsu
,
H. C.
,
Kibler
,
B.
, and
Chabchoub
,
A.
,
2017
, “
Nonconservative Higher-Order Hydrodynamic Modulation Instability
,”
Phys. Rev. E
,
96
(
2
), p.
022219
.10.1103/PhysRevE.96.022219
183.
Dudley
,
J. M.
,
Dias
,
F.
,
Erkintalo
,
M.
, and
Genty
,
G.
,
2014
, “
Instabilities, Breathers and Rogue Waves in Optics
,”
Nat. Photonics
,
8
(
10
), p.
755
.10.1038/nphoton.2014.220
184.
Närhi
,
M.
,
Wetzel
,
B.
,
Billet
,
C.
,
Toenger
,
S.
,
Sylvestre
,
T.
,
Merolla
,
J.-M.
,
Morandotti
,
R.
,
Dias
,
F.
,
Genty
,
G.
, and
Dudley
,
J. M.
,
2016
, “
Real-Time Measurements of Spontaneous Breathers and Rogue Wave Events in Optical Fibre Modulation Instability
,”
Nat. Commun.
,
7
, p.
13675
.10.1038/ncomms13675
185.
Chabchoub
,
A.
,
2016
, “
Tracking Breather Dynamics in Irregular Sea State Conditions
,”
Phys. Rev. Lett.
,
117
(
14
), p.
144103
.10.1103/PhysRevLett.117.144103
186.
Cousins
,
W.
, and
Sapsis
,
T. P.
,
2015
, “
Unsteady Evolution of Localized Unidirectional Deep-Water Wave Groups
,”
Phys. Rev. E
,
91
(
6
), p.
063204
.10.1103/PhysRevE.91.063204
187.
Cousins
,
W.
, and
Sapsis
,
T. P.
,
2016
, “
Reduced-Order Precursors of Rare Events in Unidirectional Nonlinear Water Waves
,”
J. Fluid Mech.
,
790
(
3
), pp.
368
388
.10.1017/jfm.2016.13
188.
Farazmand
,
M.
, and
Sapsis
,
T. P.
,
2017
, “
Reduced-Order Prediction of Rogue Waves in Two-Dimensional Deep-Water Waves
,”
J. Comput. Phys.
,
340
, pp.
418
434
.10.1016/j.jcp.2017.03.054
189.
Ohta
,
Y.
, and
Yang
,
J.
,
2012
, “
Rogue Waves in the Davey-Stewartson—I: Equation
,”
Phys. Rev. E
,
86
(
3
), p.
036604
.10.1103/PhysRevE.86.036604
190.
Dematteis
,
G.
,
Grafke
,
T.
, and
Vanden-Eijnden
,
E.
,
2018
, “
Rogue Waves and Large Deviations in Deep Sea
,”
Proc. Natl. Acad. Sci.
,
115
(
5
), pp.
855
860
.10.1073/pnas.1710670115
191.
Davison
,
A. C.
, and
Huser
,
R.
,
2015
, “
Statistics of Extremes
,”
Annu. Rev. Stat. Appl.
,
2
(
1
), pp.
203
235
.10.1146/annurev-statistics-010814-020133
192.
Benjamin
,
T. B.
, and
Feir
,
J. E.
,
1967
, “
The Disintegration of Wave Trains on Deep Water—Part 1: Theory
,”
J. Fluid Mech.
,
27
(
3
), pp.
417
430
.10.1017/S002211206700045X
193.
Drazen
,
A.
,
2000
,
Political Economy in Macroeconomics
,
Princeton University Press
, Princeton, NJ.
194.
Keizer
,
K.
,
Lindenberg
,
S.
, and
Steg
,
L.
,
2008
, “
The Spreading of Disorder
,”
Science
,
322
(
5908
), pp.
1681
1685
.10.1126/science.1161405
195.
Rand
,
D. G.
,
Arbesman
,
S.
, and
Christakis
,
N. A.
,
2011
, “
Dynamic Social Networks Promote Cooperation in Experiments With Humans
,”
Proc. Natl. Acad. Sci.
,
108
(
48
), pp.
19193
19198
.10.1073/pnas.1108243108
196.
Kloeden
,
P. E.
, and
Rasmussen
,
M.
,
2011
,
Nonautonomous Dynamical Systems
, Vol.
176
,
American Mathematical Society
, Providence, RI.
197.
Carvalho
,
A.
,
Langa
,
J. A.
,
Robinson
,
J.
,
2012
,
Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems
, Vol.
182
,
Springer Science & Business Media
, New York.
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