Intermittent large amplitude events are seen in the temporal evolution of a state variable of many dynamical systems. Such intermittent large events suddenly start appearing in dynamical systems at a critical value of a system parameter and continues for a range of parameter values. Three important processes of instabilities, namely, interior crisis, Pomeau–Manneville intermittency, and the breakdown of quasiperiodic motion, are most common as observed in many systems that lead to such occasional and rare transitions to large amplitude spiking events. We characterize these occasional large events as extreme events if they are larger than a statistically defined significant height. We present two exemplary systems, a single system and a coupled system, to illustrate how the instabilities work to originate extreme events and they manifest as non-trivial dynamical events. We illustrate the dynamical and statistical properties of such events.
REFERENCES
1.
D. R.
Easterling
, G. A.
Meehl
, C.
Parmesan
, S. A.
Changnon
, T. R.
Karl
, and L. O.
Mearns
, Science
289
, 2068
(2000
). 2.
Extreme Events in Nature and Society, edited by S. Albeverio, V. Jentsch, and H. Kantz (Springer, Heidelberg, 2005).
3.
K.
Lehnertz
, “Epilepsy: Extreme events in the human brain,” in Extreme Events in Nature and Society (Springer, Heidelberg, 2005), p. 123.4.
K.
Lehnertz
, J. Biol. Phys.
34
, 253
(2008
). 5.
L. E.
McPhillips
et al., Earth’s Future
6
, 441
(2018
). 6.
K. E.
Trenberth
, A.
Dai
, R. M.
Rasmussen
, and D. B.
Parsons
, Bull. Am. Met. Soc.
84
, 1205
(2003
). 7.
B. N.
Goswami
, V.
Venugopal
, D.
Sengupta
, M. S.
Madhusoodanan
, and P. K.
Xavier
, Science
314
, 1442
(2006
). 8.
N.
Akhmediev
et al., J. Optics
18
, 063001
(2016
). 9.
C.
Kharif
, E.
Pelinovsky
, and A.
Slunyaev
, Rogue Waves in the Ocean
, Advances in Geophysical and Environmental Mechanics and Mathematics (Springer
, Berlin
, 2009
).10.
K.
Dysthe
, H. E.
Krogstad
, and P.
Müller
, Annu. Rev. Fluid Mech.
40
, 287
(2008
). 11.
S. M.
Krause
, S.
Börries
, and S.
Bornholdt
, Phys. Rev. E
92
, 012815
(2015
). 12.
R.
Biggs
, S. R.
Carpenter
, and W. A.
Brock
, Proc. Natl. Acad. Sci. U.S.A.
106
, 826
(2009
). 13.
J. M.
Dudley
, F.
Dias
, M.
Erkintalo
, and G.
Genty
, Nat. Photonics
8
, 755
(2014
). 14.
H. L. D.
de S. Cavalcante
, M.
Oriá
, D.
Sornette
, E.
Ott
, and D. J.
Gauthier
, Phys. Rev. Lett.
111
, 013901
(2013
). 15.
G.
Ansmann
, R.
Karnatak
, K.
Lehnertz
, and U.
Feudel
, Phys. Rev. E
88
, 052911
(2013
). 16.
G.
Messori
et al., Bull. Am. Meteor. Soc.
99
, ES81
(2018
). 17.
M.
Farazmand
and T. P.
Sapsis
, Sci. Adv.
3
, e1701533
(2017
). 18.
C.
Mulhern
, S.
Bialonski
, and H.
Kantz
, Phys. Rev. E
91
, 012918
(2015
). 19.
M.
Farazmand
and T.
Sapsis
, Appl. Mech. Rev.
71
, 050801
(2019
) and references therein. 20.
R.
Karnatak
, H.
Kantz
, and S.
Bialonski
, Phys. Rev. E
96
, 042211
(2017
). 21.
S. L.
Kingston
, K.
Thamilamaran P. Pal
, U.
Feudel
, and S. K.
Dana
, Phys. Rev. E
96
, 052204
(2017
). 22.
C.
Bonatto
, M.
Feyereisen
, S.
Barland
, M.
Giudici
, C.
Masoller
, J.
Rios Leite
, and J. R.
Tredicce
, Phys. Rev. Lett.
107
, 053901
(2011
). 23.
N. M.
Alvarez
, S.
Borkar
, and C.
Masoller
, Eur. Phys. J. Spec. Top.
226
, 1971
(2017
);J.
Zamora-Munt
, B.
Garbin
, S.
Barland
, M.
Giudici
, and J. R.
Rios Leite
, Phys. Rev. A
87
, 035802
(2013
). 24.
J. A.
Reinoso
, J.
Zamora-Munt
, and C.
Masoller
, Phys. Rev. E
87
, 062913
(2013
). 25.
D. R.
Solli
, C.
Ropers
, P.
Koonath
, and B.
Jalali
, Nature
450
, 1054
(2007
). 26.
K.
Panajotov
, M. G.
Clerc
, and M.
Tlidi
, Eur. Phys. J. D
71
, 176
(2017
). 27.
S.
Perrone
, J.
Zamora-Munt
, R.
Vilaseca
, and C.
Masoller
, Phys. Rev. A
89
, 033804
(2014
). 28.
M. G.
Clerc
, G.
Gonzalez-Cortres
, and M.
Wilson
, Opt. Lett.
41
, 2711
(2016
). 29.
A.
Chabchoub
, N.
Hoffmann
, and N.
Akhmediev
, J. Geophys. Res. Oceans
117
, C00J02
, https://doi.org/10.1029/2011JC007636 (2012
);A.
Chabchoub
, N.
Hoffmann
, M.
Onorato
, and N.
Akhmediev
, Phys. Rev. X
2
, 011015
(2012
).30.
A. N.
Pisarchik
, R.
Jaimes-Reategui
, R.
Sevilla-Escoboza
, G.
Huerta-Cuellar
, and M.
Taki
, Phys. Rev. Lett.
107
, 274101
(2011
). 31.
S.
Kumarasamy
and A. N.
Pisarchik
, Phys. Rev. E
98
, 032203
(2018
). 32.
R.
Karnatak
, G.
Ansmann
, U.
Feudel
, and K.
Lehnertz
, Phys. Rev. E
90
, 022917
(2014
). 33.
E. J.
Doedel
and C. L.
Pando L.
, Phys. Rev. E
100
, 052204
(2019
). 34.
F.
Rogister
and R.
Roy
, Phys. Rev. Lett.
98
, 104101
(2007
). 35.
A.
Mishra
, S.
Saha
, M.
Vigneshwaran
, T.
Kapitaniak
, and S. K.
Dana
, Phys. Rev. E
97
, 062311
(2018
). 36.
A.
Roy
, A.
Mishra
, D.
Ghosh
, T.
Kapitaniak
, S. K.
Dana
, and C. R.
Hens
, Phys. Rev. E
101
, 032209
(2020
). 37.
R.
Suresh
and V. K.
Chandrasekar
, Phys. Rev. E
98
, 052211
(2018
). 38.
V.
Kishore
, A. R.
Sonawane
, and M. S.
Santhanam
, Phys. Rev. E
88
, 014801
(2013
). 39.
M.
Farazmand
and T.
Sapsis
, Phys. Rev. E
100
, 033110
(2019
). 40.
P. P.
Galuzio
, R. L.
Viana
, and S. R.
Lopez
, Phys. Rev. E
89
, 040901
(2014
). 41.
A.
Roy
, S.
Rakshit
, D.
Ghosh
, and S. K.
Dana
, Chaos
29
, 043131
(2019
). 42.
T.
Sapsis
, Phil. Trans. R. Soc. A
376
, 20170133
(2018
). 43.
A.
Saha
and U. Feudel, Eur. Phys. J. Spec. Top.
227
, 1205
(2018
). 44.
N.
Blackbeard
, S.
Wieczorek
, H.
Erzgräber
, and P. S.
Dutta
, Physica D
286–287
, 43
(2014
). 45.
S. C.
Venkataramani
, B. R.
Hunt
, E.
Ott
, D. J.
Gauthier
, and J. C.
Bienfang
, Phys. Rev. Lett.
77
, 5361
(1996
);
[PubMed]
S. C.
Venkataramani
, B. R.
Hunt
, and E.
Ott
, Phys. Rev. E
54
, 1346
(1996
). 46.
P.
Ashwin
, J.
Buescu
, and I.
Stewart
, Nonlinearity
9
, 703
(1996
);P.
Ashwin
, J.
Buescu
, and I.
Stewart
, Phys. Lett. A
193
, 126
(1994
). 47.
S.
Wieczorek
, P.
Ashwin
, C.
Luke
, and P.
Cox
, Proc. R. Soc. Lond. A
467
, 1243
(2011
). 48.
C.
Grebogi
, E.
Ott
, F.
Romeiras
, and J. A.
Yorke
, Phys. Rev. A
36
, 5365
(1987
);C.
Grebogi
, E.
Ott
, and J. A.
Yorke
, Physica D
7
, 181
(1983
). 49.
A.
Saha
and U.
Feudel
, Phys. Rev. E
95
, 062219
(2017
). 50.
A.
Roy
, S.
Rakshit
, G.
Basak
, D.
Ghosh
, and S. K.
Dana
, “Understanding origin of extreme events in El-Niño southern oscillation
,” Phys. Rev. E
(to be published).51.
Y.
Pomeau
and P.
Manneville
, Commun. Math. Phys.
74
, 189
(1980
). 52.
V.
Nair
, G.
Thampi
, and R. I.
Sujith
, J. Fluid Mech.
756
, 470
(2014
). 53.
C.
Nicolis
and G.
Nicolis
, Eur. Phys. Lett.
80
, 40003
(2007
). 54.
J. L.
Hindmarsh
and R. M.
Rose
, Proc. R. Soc. Lond. B Biol. Sci.
221
, 87
(1984
).55.
A.
Mishra
, C.
Hens
, M.
Bose
, P. K.
Roy
, and S. K.
Dana
, Phys. Rev. E
92
, 062920
(2015
). 56.
J.
Guckeheimer
and P.
Holmes
, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields
(Springer-Verlag
, New York
, 1983
).© 2020 Author(s).
2020
Author(s)
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