Abstract
It is natural to view climate as a slow motion governed by a set of equations depending on weather considered as a fast motion governed by another set of equations. This leads to the averaging setup which is discussed in this paper.
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References
D.B. Anosov, Averaging in systems of ordinary differential equations with fast oscillating solutions, Izv. Acad. Nauk SSSR Ser. Mat, 24 (1960), 731–742 (Russian).
L. Arnold, Hasselmann’s program revisited: the analysis of stochasticity in deterministic climate models, this Proceedings.
D.K. Arrowsmith and C.M. Place, An Introduction to Dynamical Systems, (1990), Cambridge Univ. Press, Cambridge.
M. Brin and M. Freidlin, On stochastic behavior of perturbed Hamiltonian systems, Ergod. Th. & Dynam. Sys., 20 (2000), 55–76.
N.N. Bogolyubov and Yu.A. Mitropol’skii, Asymptotic Methods in the Theory of Nonlinear Oscillations, (1961), Hindustan Publ. Co., Delhi.
G. Contreras, Regularity of topological and metric entropy of hyperbolic flows, Math. Z., 210 (1992), 97–111.
M.I. Freidlin, The averaging principle and theorems on large deviations, Russ. Math. Surv., No. 5, 33 (1978), 107–160.
M.I. Preidlin and A.D. Wentzell, Random Perturbations of Dynamical Systems, 2nd ed., (1998), Springer-Verlag, New York.
G. Gallavotti, Chaotic hypotesis and universal large deviations properties, Doc. Math. J. DMV, Extra Volume ICM 1998, I(1998), 205–233.
K. Hasselmann, Stochastic climate models, Part I. Theory, Tellus, 28 (1976), 473–485.
D. Husemoller, Fibre Bundles, 3rd ed., (1994), Springer-Verlag, New York.
R.Z. Khasminskii, On stochastic processes defined by differential equations with a small parameter, Th. Probab. Appl., 11 (1966), 211–228.
R.Z. Khasminskii, On the averaging principle for Itô stochastic differential equations, Kibernetika (Prague), 4 (1968), 260–279 (Russian).
Yu. Kifer, Principal eigenvalues, topological pressure, and stochastic stability of equilibrium states, Israel J. Math., 70 (1990), 1–47.
Yu. Kifer, Averaging in dynamical systems and large deviations, Invent. Math., 110 (1992), 337–370.
Yu. Kifer, Limit theorems in averaging for dynamical systems, Ergod. Th.& Dynam. Sys., 15 (1995), 1143–1172.
A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, (1995), Cambridge Univ. Press, Cambridge.
M.A. Krasnoselskii and S.G. Krein, On the averaging principle in nonlinear mechanics, Uspekhi Mat. Nauk, No. 3, 10 (1955), 147–152 (Russian).
P. Lochak and C. Meunier, Multiple Averaging for Classical Systems, (1988), Springer-Verlag, New York.
E. Pardoux and A.Yu. Veretennikov, On Poisson equation and diffusion approximationll, Preprint, (2000).
J.A. Sanders and F.Verhurst, Averaging Methods in Nonlinear Dynamical Systems, (1985), Springer-Verlag, Berlin.
A.Yu. Veretennikov, On the averaging principle for systems of stochastic differential equations, Math. USSR Sbornik, 69 (1991), 271–284.
A.Yu. Veretennikov, On large deviations in the averaging principle for SDEs with “full dependence”, Ann. Probab., 27 (1999), 284–296.
S. Waddington, Large deviations asymptotics for Anosov flows, Ann. Inst. H.Poincare (Ser. Anal. Non Lineaire), 13 (1996), 445–484.
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Kifer, Y. (2001). Averaging and climate models. In: Imkeller, P., von Storch, JS. (eds) Stochastic Climate Models. Progress in Probability, vol 49. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8287-3_7
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DOI: https://doi.org/10.1007/978-3-0348-8287-3_7
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