Fascination of chaos

This review introduces most of the concepts used in the study of chaotic phenomena in nonlinear systems and has as its objective to summarize the current understanding of results from the theory of chaotic dynamical systems and to describe the original ideas underlying the study of deterministic chaos. The presentation relies on informal analysis, with abstract mathematical ideas visualized geometrically or by examples from physics. Hyperbolic dynamics, homoclinic trajectories and tangencies, wild hyperbolic sets, and different types of attractors which appear in dynamical systems are considered. The key aspects of ergodic theory are discussed, and the basic statistical properties of chaotic dynamical systems are described. The fundamental difference between stochastic dynamics and deterministic chaos is explained. The review concludes with an investigation of the possibility of studying complex systems on the basis of the analysis of registered signals, i.e., the generated time series.

PACS: 02.50.−r, 05.45.−a, 47.52.+j (all)
DOI: 10.3367/UFNe.0180.201012d.1305
URL: https://ufn.ru/en/articles/2010/12/d/
Web of Science

000288495700004
Scopus

2-s2.0-79955613299
Citation: Loskutov A "Fascination of chaos" Phys. Usp. 53 1257–1280 (2010)

BibTexBibNote ® (generic)BibNote ® (RIS)MedlineRefWorks

Оригинал: Лоскутов А Ю «Очарование хаоса» УФН 180 1305–1329 (2010); DOI: 10.3367/UFNr.0180.201012d.1305

References (182) Cited by (37) Similar articles (20) ↓

M.V. Kalashnik, M.V. Kurgansky, O.G. Chkhetiani “Baroclinic instability in geophysical fluid dynamics” Phys. Usp. 65 1039–1070 (2022)
S.P. Kuznetsov “Dynamical chaos and uniformly hyperbolic attractors: from mathematics to physics” Phys. Usp. 54 119–144 (2011)
K.V. Koshel, S.V. Prants “Chaotic advection in the ocean” Phys. Usp. 49 1151–1178 (2006)
V.S. Anishchenko, T.E. Vadivasova et al “Statistical properties of dynamical chaos” Phys. Usp. 48 151–166 (2005)
O.G. Onishchenko, O.A. Pokhotelov et al “Structure and dynamics of concentrated mesoscale vortices in planetary atmospheres” Phys. Usp. 63 683–697 (2020)
B.M. Smirnov “Electrical cycle in the Earth’s atmosphere” Phys. Usp. 57 1041–1062 (2014)
A.N. Vulfson, O.O. Borodin “The system of convective thermals as a generalized ensemble of Brownian particles” Phys. Usp. 59 109–120 (2016)
M.I. Rabinovich, M.M. Sushchik “The regular and chaotic dynamics of structures in fluid flows” Sov. Phys. Usp. 33 (1) 1–35 (1990)
V.S. Anishchenko, A.B. Neiman et al “Stochastic resonance: noise-enhanced order” Phys. Usp. 42 7–36 (1999)
O.G. Onishchenko, O.A. Pokhotelov, N.M. Astaf’eva “Generation of large-scale eddies and zonal winds in planetary atmospheres” Phys. Usp. 51 577–589 (2008)
V.M. Fedorov “Problems of parameterization of the radiation block in physical and mathematical climate models and the possibility of their solution” Phys. Usp. 66 914–930 (2023)
G.I. Strelkova, V.S. Anishchenko “Spatio-temporal structures in ensembles of coupled chaotic systems” Phys. Usp. 63 145–161 (2020)
F.V. Dolzhanskii, V.A. Krymov, D.Yu. Manin “Stability and vortex structures of quasi-two-dimensional shear flows” Sov. Phys. Usp. 33 (7) 495–520 (1990)
A.L. Virovlyansky, D.V. Makarov, S.V. Prants “Ray and wave chaos in underwater acoustic waveguides” Phys. Usp. 55 18–46 (2012)
A.A. Koronovskii, O.I. Moskalenko, A.E. Hramov “On the use of chaotic synchronization for secure communication” Phys. Usp. 52 1213–1238 (2009)
V.S. Anishchenko, S.V. Astakhov “Poincaré recurrence theory and its applications to nonlinear physics” Phys. Usp. 56 955–972 (2013)
V.P. Budaev, S.P. Savin, L.M. Zelenyi “Investigation of intermittency and generalized self-similarity of turbulent boundary layers in laboratory and magnetospheric plasmas: towards a quantitative definition of plasma transport features” Phys. Usp. 54 875–918 (2011)
V.I. Klyatskin “Integral characteristics: a key to understanding structure formation in stochastic dynamic systems” Phys. Usp. 54 441–464 (2011)
V.V. Zosimov, L.M. Lyamshev “Fractals in wave processes” Phys. Usp. 38 347–384 (1995)
Ya.B. Zel’dovich, S.A. Molchanov et al “Intermittency in random media” Sov. Phys. Usp. 30 353–369 (1987)

The list is formed automatically.