This paper introduces the notion of robust extremes in deterministic chaotic systems, presents initial theoretical results, and outlines associated inferential techniques. A chaotic deterministic system is said to exhibit robust extremes under a given observable when the associated statistics of extreme values depend smoothly on the system’s control parameters. Robust extremes are here illustrated numerically for the flow of the Lorenz model [E. N. Lorenz, J. Atmos. Sci. 20, 130 (1963)]. Robustness of extremes is proved for one-dimensional Lorenz maps with two distinct types of observables for which conditions guaranteeing robust extremes are formulated explicitly. Two applications are shown: improving the precision of the statistical estimator for extreme value distributions and predicting future extremes in nonstationary systems. For the latter, extreme wind speeds are examined in a simple quasigeostrophic model with a robust chaotic attractor subject to nonstationary forcing.
REFERENCES
1.
J.
Guckenheimer
and P.
Holmes
, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields
(Springer-Verlag
, Berlin
, 1983
).2.
H. W.
Broer
, Bull., New Ser., Am. Math. Soc.
41
, 507
(2004
).3.
L. D.
Landau
and E. M.
Lifshitz
, Course of Theoretical Physics. Vol. 5: Statistical Physics
, translated from the Russian by J. B. Sykes and M. J. Kearsley, second revised and enlarged edition (Pergamon
, New York
, 1968
).4.
C.
Grebogi
, E.
Ott
, and J. A.
Yorke
, Phys. Rev. Lett.
48
, 1507
(1982
).5.
J.
Palis
and F.
Takens
, Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations
(Cambridge University Press
, Cambridge
, 1993
).6.
S.
Banerjee
, J. A.
Yorke
, and C.
Grebogi
, Phys. Rev. Lett.
80
, 3049
(1998
).7.
A.
Potapov
and M. K.
Ali
, Phys. Lett. A
277
, 310
(2000
).8.
M.
Andrecut
and M. K.
Ali
, Phys. Rev. E
64
, 025203
(R) (2001
).9.
P.
Kowalczyk
, Nonlinearity
18
, 485
(2005
).10.
Z.
Elhadj
and J.
Sprott
, Fron. Phys. China
3
, 195
(2008
).11.
12.
E.
Lorenz
, J. Atmos. Sci.
20
, 130
(1963
).13.
C. A.
Morales
, M. J.
Pacifico
, and E. R.
Pujals
, Proc. Am. Math. Soc.
127
, 3393
(1999
).14.
J. F.
Alves
, M.
Carvalho
, and J. M.
Freitas
, “Statistical stability for Hénon maps of the Benedicks-Carleson type
,” Ann. Inst. Henri Poincare, Anal. Non Lineaire
(to be published), e-print arXiv:math/0610602.15.
J. M.
Freitas
and M.
Todd
, Nonlinearity
22
, 259
(2009
).16.
E.
Castillo
, Extreme Value Theory in Engineering
, Statistical Modeling and Decision Science
(Academic
, New York
, 1988
).17.
P.
Embrechts
, C.
Klüppelberg
, and T.
Mikosch
, Modelling Extremal Events for Insurance and Finance
, Applications of Mathematics (New York)
, Vol. 33
(Springer-Verlag
, Berlin
, 1997
).18.
S.
Coles
, An Introduction to Statistical Modeling of Extreme Values
, Springer Series in Statistics
(Springer
, New York
, 2001
).19.
J.
Beirlant
, Y.
Goegebeur
, J.
Teugels
, and J.
Segers
, Statistics of Extremes: Theory and Applications
(Wiley
, New York
, 2004
).20.
P.
Collet
, Ergod. Theory Dyn. Syst.
21
, 401
(2001
).21.
M.
Holland
, M.
Nicol
, and A.
Török
, “Extreme value distributions for non-uniformly hyperbolic dynamical systems
” (unpublished).22.
A. C. M.
Freitas
and J. M.
Freitas
, Ergod. Theory Dyn. Syst.
28
, 1117
(2008
).23.
A. C. M.
Freitas
, J. M.
Freitas
, and M.
Todd
, “Hitting time statistics and extreme value theory
,” Probab. Theory Relat. Fields
(to be published), e-print arXiv:0804.2887v2.24.
M.
Felici
, V.
Lucarini
, A.
Speranza
, and R.
Vitolo
, J. Atmos. Sci.
64
, 2159
(2007
).25.
N. H.
Bingham
, J. Comput. Appl. Math.
200
, 357
(2007
).26.
M. R.
Leadbetter
, Z. Wahrscheinlichkeitstheor. Verwandte Geb.
65
, 291
(1983
).27.
C.
Sparrow
, The Lorenz equations: bifurcations, chaos, and strange attractors
, Applied Mathematical Sciences
, Vol. 41
(Springer-Verlag
, New York
, 1982
).29.
M.
Felici
, V.
Lucarini
, A.
Speranza
, and R.
Vitolo
, J. Atmos. Sci.
64
, 2137
(2007
).30.
V.
Lucarini
, A.
Speranza
, and R.
Vitolo
, Physica D
234
, 105
(2007
).31.
E.
Kriegler
, J. W.
Hall
, H.
Held
, R.
Dawson
, and H. J.
Schellnhuber
, Proc. Natl. Acad. Sci. U.S.A.
106
, 5041
(2009
).32.
L. -S.
Young
, J. Stat. Phys.
108
, 733
(2002
).33.
C.
Gupta
, M.
Holland
, and M.
Nicol
, “Extreme value theory for Lorenz maps and flows, Henon-like diffeomorphisms and a class of hyperbolic systems with singularities
” (unpublished).34.
A. C. M.
Freitas
and J. M.
Freitas
, Stat. Probab. Lett.
78
, 1088
(2008
).35.
L.
Mendoza
, Mathematical Notes, No. 100 (Spanish)
(Univ. de Los Andes
, Mérida
, 1989
), pp. 91
–101
.36.
V.
Araujo
, M. J.
Pacifico
, E. R.
Pujals
, and M.
Viana
, Trans. Am. Math. Soc.
361
, 2431
(2009
).37.
S.
Luzzatto
and W.
Tucker
, Publ. Math., Inst. Hautes Etud. Sci.
89
, 179
(1999
).38.
S.
Luzzatto
, I.
Melbourne
, and F.
Paccaut
, Commun. Math. Phys.
260
, 393
(2005
).39.
The algorithm described in Broer et al. (Ref. 52) has been used.
40.
D.
Ruelle
, Commun. Math. Phys.
187
, 227
(1997
).41.
D.
Ruelle
, Ergod. Theory Dyn. Syst.
28
, 613
(2008
).42.
G.
Gallavotti
and E. G. D.
Cohen
, J. Stat. Phys.
80
, 931
(1995
).43.
F.
Bonetto
, G.
Gallavotti
, A.
Giuliani
, and F.
Zamponi
, J. Stat. Phys.
123
, 39
(2006
).44.
G.
Gallavotti
, Foundations of Fluid Dynamics
(Springer-Verlag
, Berlin
, 2002
).45.
D. J.
Albers
, J. C.
Sprott
, and J. P.
Crutchfield
, Phys. Rev. E
74
, 057201
(2006
).46.
J. M.
Murphy
, B. B. B.
Booth
, M.
Collins
, G. R.
Harris
, D. M. H.
Sexton
, and M. J.
Webb
, Philos. Trans. R. Soc. London, Ser. A
365
, 1993
(2007
).47.
R. A.
Rigby
and D. M.
Stasinopoulos
, J. R. Stat. Soc., Ser. C, Appl. Stat.
54
, 507
(2005
).48.
49.
T. J.
Hastie
, in Statistical Models in S
, edited by J. M.
Chambers
and T. J.
Hastie
(Brooks-Cole
, Belmont
, 1991
), pp. 309
–376
.50.
J. F.
Alves
, C.
Bonatti
, and M.
Viana
, Invent. Math.
140
, 351
(2000
).51.
52.
H.
Broer
, C.
Simó
, and R.
Vitolo
, Nonlinearity
15
, 1205
(2002
).© 2009 American Institute of Physics.
2009
American Institute of Physics
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