Abstract
For a dynamical system with slow and fast variables the popular method of averaging enables one to derive an equation for the slow variables alone whose solution approximates the original slow motion on a finite time interval.
If the fast variables are sufficiently “random” the error term in the averaging procedure is described by a central limit theorem, i.e., the scaled error is Gaussian in the weak limit and satisfies a linear SDE (linear diffusion, or Gaussian approximation).
We will present an approximation of the slow motion by the solution of a nonlinear SDE (in meteorology known as Hasselmann’s equation) which was recently proved by Yuri Kifer. Although this nonlinear diffusion approximation holds in general, as the previous one’s, only on a finite time interval, it is at least in principle capable of correctly describing important long-term qualitative features of the slow motion.
We present several examples which support the usefulness of the nonlinear diffusion approximation, including the Lorenz-Maas model from climatology.
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References
L. Arnold, Random dynamical systems. Springer-Verlag, Berlin Heidelberg New York, 1998.
L. Arnold, Hasselmann’s program revisited: the analysis of stochasticity in deterministic climate models. In P. Imkeller and J.-S. von Storch, editors, Stochastic Climate Models, volume 49 of Progress in Probability, pp. 141–157, Birkhäuser, Boston Basel Stuttgart, 2001.
L. Arnold, P. Imkeller, and Y. Wu, Reduction of deterministic coupled atmosphere-ocean models to stochastic ocean models: A numerical case study of the Lorenz-Maas system, Report, Institut für Dynamische Systeme, Universität Bremen, 2002.
I. Berkes and W. Philipp, “Approximation theorems for independent and weakly dependent random vectors,” Ann. Probab, vol. 7, pp. 29–54, 1979.
N. Bogolyubov and Y. Mitropolskii, Asymptotic methods in the theory of nonlinear oscillations, Gordon and Breach, NY, 1961.
H. Crauel and F. Flandoli, “Additive noise destroys a pitchfork bifurcation,” Journal of Dynamics and Differential Equations, vol. 10, pp. 259–274, 1998.
R. Dudley, “Distances of probability measure and random variables,” Ann. Math. Statist., vol. 39, pp. 1563–1572, 1968.
M. Freidlin, “The averaging principle and theorems on large deviations,” Russian Math. Surveys, vol. 33, pp. 117–176, 1978.
M. Freidlin and A. Wentzell, Random perturbations of dynamical systems. Springer-Verlag, Berlin, Heidelberg, New York, 1998. Second edition.
K. Hasselmann, “Stochastic climate models: Part I. Theory,” Tellus, vol. 28, pp. 473–485, 1976.
I. A. Ibragimov and Y. V. Linnik, Independent and stationary sequences of random variables. Wolters-Noordhoff, Groningen, 1971.
P. Imkeller and A. Monahan, editors. Stochastic Climate Models, volume 2 of Stochastics and Dynamics, World Scientific, Singapore, 2002. Special Issue.
P. Imkeller and J.-S. von Storch, editors. Stochastic Climate Models, volume 49 of Progress in Probability, Birkhäuser, Boston Basel Stuttgart, 2001.
W. Just, H. Kantz, C. Rödenbeck, and M. Helm, “Stochastic modelling: replacing fast degrees of freedom by noise,” J Phys. A: Math. Gen., vol. 34, pp. 3199–3213, 2001.
H. Kantz, W. Just, and N. Baba, “Replacing fast chaotic degrees of freedom by noise: a formally exact result,” Preprint, 2002.
R. Z. Khasminskii, “A limit theorem for solutions of differential equations with random right-hand side,” Theory of Probability and its Applications, vol. 11, pp. 390–406, 1966.
R. Z. Khasminskii, “On stochastic processes defined by differential equations with a small parameter,” Theory of Probability and its Applications, vol. 11, pp. 211–228, 1966.
R. Z. Khasminskii, “On the principle of averaging for Itô stochastic differential equations,” (in Russian). Kybernetika, vol. 4, pp. 260–279, 1968.
Y. Kifer, “Large deviations in dynamical systems and stochastic processes,” Trans. Amer. Math. Soc, vol. 321, pp. 505–524, 1990.
Y. Kifer, “Averaging in dynamical systems and large deviations,” Invent, math., vol. 110, pp. 337–370, 1992.
Y Kifer, “Limit theorems in averaging for dynamical systems,” Ergodic Theory and Dynamical Systems, vol. 15, pp. 1143–1172, 1995.
Y. Kifer, Averaging and climate models. In P. Imkeller and J.-S. von Storch, editors, Stochastic Climate Models, volume 49 of Progress in Probability, pages 171–188, Birkhäuser, Boston Basel Stuttgart, 2001.
Y Kifer, L 2 diffusion approximation for slow motion in averaging. Preprint, Institute of Mathematics, Hebrew University, 2002.
J. Kuelbs and W. Philipp, “Almost sure invariance principles for partial sums of mixing B-valued random variables,” Ann. Probab., vol. 8, pp. 1003–1036, 1980.
R. Liptser and A. Shiryayev, Theory of martingales, Kluwer, Dordrecht, 1989.
E. Lorenz, “Irregularity: a fundamental property of the atmosphere,” Tellus, vol. 36A, pp. 98–110, 1984.
L. Maas, “A simple model for the three-dimensional, thermally and wind-driven ocean circulation,” Tellus, vol. 46A, pp. 671–680, 1994.
A. Majda, I. Timofeyev, and E. Vanden-Eijnden, “A mathematical framework for stochastic climate models,” Communications on Pure and Applied Mathematics, vol. 54, pp. 891–974, 2001.
A. Majda, I. Timofeyev, and E. Vanden-Eijnden, A priori tests of a stochastic mode reduction strategy, preprint, Courant Institute of Mathematical Sciences, New York University, 2002.
G. Papanicolaou, D. Stroock, and S. R. S. Varadhan. Martingale approach to some limit theorems. Duke University Mathematics Series III, 1977.
W. Philipp and W. Stout, Almost sure invariance principles for partial sums of weakly dependent random variables, Mem. Amer. Math. Soc. 161, 1975.
Y. A. Rozanov, Stationary random processes, Holden-Day, San Francisco, 1967.
J. A. Sanders and F. Verhulst, Averaging methods in nonlinear dynamical systems, Springer-Verlag, Berlin, Heidelberg, New York, 1985.
V. Strassen, “The existence of probability measures with given marginals,” Ann. Math. Statist., vol. 36, pp. 423–439, 1965.
L. van Veen, The Maas-Lorenz model. Manuscript, Department of Mathematics, Universiteit Utrecht, 2001.
L. van Veen, Time scale interaction in low-order climate models, PhD thesis, Department of Mathematics, Universiteit Utrecht, 2002.
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Arnold, L. (2003). Linear and Nonlinear Diffusion Approximation of the Slow Motion in Systems with Two Time Scales. In: Namachchivaya, N.S., Lin, Y.K. (eds) IUTAM Symposium on Nonlinear Stochastic Dynamics. Solid Mechanics and Its Applications, vol 110. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0179-3_1
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