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/
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000258187900005
Scopus

2-s2.0-49249111465
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2008PhyU...51..395K
Citation: Klyatskin V I "Statistical topography and Lyapunov exponents in stochastic dynamical systems" Phys. Usp. 51 395–407 (2008)

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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) ↓

V.I. Klyatskin, A.I. Saichev “Statistical and dynamic localization of plane waves in randomly layered media” Sov. Phys. Usp. 35 (3) 231–247 (1992)
V.I. Klyatskin, K.V. Koshel’ “Simple example of the development of cluster structure of a passive tracer field in random flows” Phys. Usp. 43 717–723 (2000)
O.G. Bakunin “Correlation and percolation properties of turbulent diffusion” Phys. Usp. 46 733–744 (2003)
V.I. Klyatskin “Statistical description of the diffusion of a passive tracer in a random velocity field” Phys. Usp. 37 501–513 (1994)
S.N. Gordienko “Irreversibility and the probabilistic treatment of the dynamics of classical particles” Phys. Usp. 42 573–590 (1999)
V.I. Tatarskii “Example of the description of dissipative processes in terms of reversible dynamic equations and some comments on the fluctuation-dissipation theorem” Sov. Phys. Usp. 30 134–152 (1987)
A.V. Borisov, A.O. Kazakov, S.P. Kuznetsov “Nonlinear dynamics of the rattleback: a nonholonomic model” Phys. Usp. 57 453–460 (2014)
A. Loskutov “Dynamical chaos: systems of classical mechanics” Phys. Usp. 50 939–964 (2007)
V.V. Brazhkin “Why does statistical mechanics 'work' in condensed matter?” Phys. Usp. 64 1049–1057 (2021)
V.M. Rozenbaum, I.V. Shapochkina, L.I. Trakhtenberg “Green's function method in the theory of Brownian motors” Phys. Usp. 62 496–509 (2019)
G.S. Golitsyn “A N Kolmogorov's 1934 paper is the basis for explaining the statistics of natural phenomena of the macrocosm” Phys. Usp. 67 80–90 (2024)
A.N. Pavlov, V.S. Anishchenko “Multifractal analysis of complex signals” Phys. Usp. 50 819–834 (2007)
A.V. Borisov, I.S. Mamaev “Strange attractors in rattleback dynamics” Phys. Usp. 46 393–403 (2003)
V.M. Rozenbaum, I.V. Shapochkina, L.I. Trakhtenberg “Quantum particle in a V-shaped well of arbitrary asymmetry. Brownian motors” Phys. Usp. 67 1046–1055 (2024)
M.V. Kondrin, Yu.B. Lebed’ et al “On the problem of thermodynamic fluctuations in computer simulations” Phys. Usp. 68 941–946 (2025)
P.S. Landa, Ya.B. Duboshinskii “Self-oscillatory systems with high-frequency energy sources” Sov. Phys. Usp. 32 723–731 (1989)
E.N. Rumanov “Critical phenomena far from equilibrium” Phys. Usp. 56 93–102 (2013)
O.V. Rudenko “Nonlinear dynamics of quadratically cubic systems” Phys. Usp. 56 683–690 (2013)
G.N. Bochkov, Yu.E. Kuzovlev “Fluctuation-dissipation relations: achievements and misunderstandings” Phys. Usp. 56 590–602 (2013)
E.P. Zemskov “Turing patterns and Newell—Whitehead—Segel amplitude equation” Phys. Usp. 57 1035–1037 (2014)

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