Statistical topography and Lyapunov exponents in stochastic dynamical systems

This article discusses the relationship between the statistical description of stochastic dynamical systems based on the ideas of statistical topography and the traditional analysis of Lyapunov stability of dynamical systems with the use of the Lyapunov characteristic indices (Lyapunov exponents). As an illustration, some coherent phenomena are considered that occur with a probability of unity, i.e., in almost all realizations of the stochastic systems. Among such phenomena are the diffusion and clustering of a passive tracer in random hydrodynamic flows, the dynamic localization of plane waves in layered random media, and the emergence of caustic patterns of the wave field in multidimensional random media.

PACS: 05.40.−a, 05.45.−a, 42.25.Dd, 46.65.+g, 47.27.eb (all)
DOI: 10.1070/PU2008v051n04ABEH006450
URL: https://ufn.ru/en/articles/2008/4/e/
Web of Science

000258187900005
Scopus

2-s2.0-49249111465
ADS NASA

2008PhyU...51..395K
Citation: Klyatskin V I "Statistical topography and Lyapunov exponents in stochastic dynamical systems" Phys. Usp. 51 395–407 (2008)

BibTexBibNote ® (generic)BibNote ® (RIS)MedlineRefWorks

Оригинал: Кляцкин В И «Статистическая топография и ляпуновские экспоненты в динамических стохастических системах» УФН 178 419–431 (2008); DOI: 10.3367/UFNr.0178.200804e.0419

References (26) ↓ Cited by (17) Similar articles (20)

Klyatskin V I, Gurarii D Usp. Fiz. Nauk 169 171 (1999); Klyatskin V I, Gurarie D Phys. Usp. 42 165 (1999)
Klyatskin V I, v sb. Nelineinye Volny 2004 (Pod red. A V Gaponova-Grekhova, V I Nekorkina) (Nizhnii Novgorod: IPF RAN, 2005) p. 256
Klyatskin V I Stokhasticheskie Uravneniya Glazami Fizika (M.: Fizmatlit, 2001)
Klyatskin V I Dinamika Stokhasticheskikh Sistem (M.: Fizmatlit, 2002)
Klyatskin V I Diffuziya i Klasterizatsiya Passivnoi Primesi v Sluchainykh Gidrodinamicheskikh Potokakh (M.: Fizmatlit, 2005)
Isichenko M B Rev. Mod. Phys. 64 961 (1992)
Vinogradov I M (Gl. red.) Matematicheskaya Entsiklopediya T. 3 (M.: Sovetskaya entsiklopediya, 1982)
Mehlig B, Wilkinson M Phys. Rev. Lett. 92 250602 (2004)
Wilkinson M, Mehlig B Europhys. Lett. 71 186 (2005)
Saichev A I, Woyczynski W A Stochastic Models In Geosystems (The IMA Volumes in Math. and Its Applications, Vol. 85, Eds S A Molchanov, W A Woyczynski) (New York: Springer, 1997) p. 359
Zirbel C L, Cinlar E, in Stochastic Models In Geosystems (The IMA Volumes in Math. and Its Applications, Vol. 85, Eds S A Molchanov, W A Woyczynski) (New York: Springer, 1997) p. 459
Aref Kh Nelineinaya Dinamika 2 111 (2006)
Klyatskin V I, Koshel’ K V Usp. Fiz. Nauk 170 771 (2000); Klyatskin V I, Koshel’ K V Phys. Usp. 43 717 (2000)
Lifshits I M, Gredeskul S A, Pastur L A Vvedenie v Teoriyu Neuporyadochennykh Sistem (M.: Nauka, 1982); Translated into English, Lifshits I M, Gredeskul S A, Pastur L A Introduction To The Theory Of Disordered Systems (New York: Wiley, 1988)
Klyatskin V I, Saichev A I Usp. Fiz. Nauk 162 (3) 161 (1992); Klyatskin V I, Saichev A I Sov. Phys. Usp. 35 231 (1992)
Maxey M R, Riley J J Phys. Fluids 26 883 (1983)
Lamb H Hydrodynamics (New York: Dover Publ., 1945); Lemb G Gidrodinamika (M.: Gostekhizdat, 1947)
Landau L D, Lifshits E M Gidrodinamika (M.: Nauka, 1986); Translated into English, Landau L D, Lifshitz E M Fluid Mechanics (Oxford: Pergamon Press, 1987)
Tatarskii V I Rasprostranenie Voln v Turbulentnoi Atmosfere (M.: Nauka, 1967)
Rytov S M, Kravtsov Yu A, Tatarskii V I Vvedenie v Statisticheskuyu Radiofiziku T. 1, 2 (M.: Nauka, 1977, 1978); Translated into English, Rytov S M, Kravtsov Yu A, Tatarskii V I Principles Of Statistical Radiophysics (Berlin: Springer-Verlag, 1987-1989)
Klyatskin V I Usp. Fiz. Nauk 174 177 (2004); Klyatskin V I Phys. Usp. 47 169 (2004)
Gurvich A S, Kallistratova M A, Martvel’ F E Izv. Vuzov. Radiofiz. 20 1020 (1977); Gurvich A S, Kallistratova M A, Martvel E E Radiophys. Quantum Electron. 20 705 (1978)
Kravtsov Yu A Rep. Prog. Phys. 55 39 (1992)
Flatté S M, Wang G-Y, Martin J J. Opt. Soc. Am. A 10 2363 (1993)
Flatté S M, Bracher C, Wang G-Y J. Opt. Soc. Am. A 11 2080 (1994)
Dashen R J. Math. Phys. 20 894 (1979); Dashen R Opt. Lett. 9 110 (1984)