Abstract:
This paper contains some rigorous mathematical results related to the theory of development of turbulence in the sense of Landau. In diverse areas of the natural sciences, specific examples of non-linear dynamical systems are considered (including E. Hopf's classical example) whose attractors turn out to be invariant tori of arbitrarily high dimension under an appropriate change of parameters. The investigation of these examples enables us to give a rigorous meaning to the notion of a ‘turbulent attractor’ in some cases and to reveal the main properties of such an attractor, notable among which are its fractal property and its infinite dimensionality.
Citation:
A. Yu. Kolesov, N. Kh. Rozov, V. A. Sadovnichii, “Mathematical aspects of the theory of development of turbulence in the sense of Landau”, Russian Math. Surveys, 63:2 (2008), 221–282
\Bibitem{KolRozSad08}
\by A.~Yu.~Kolesov, N.~Kh.~Rozov, V.~A.~Sadovnichii
\paper Mathematical aspects of the theory of development of turbulence in the sense of Landau
\jour Russian Math. Surveys
\yr 2008
\vol 63
\issue 2
\pages 221--282
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\crossref{https://doi.org/10.1070/RM2008v063n02ABEH004515}
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Linking options:
https://www.mathnet.ru/eng/rm9171
https://doi.org/10.1070/RM2008v063n02ABEH004515
https://www.mathnet.ru/eng/rm/v63/i2/p21
This publication is cited in the following 11 articles:
Kanguzhin B.E., “On a Model of the Generation of Turbulence”, Chaos Solitons Fractals, 150 (2021), 111099
S. D. Glyzin, “Dimensional Characteristics of Diffusion Chaos”, Model. anal. inf. sist., 20:1 (2015), 30
S. D. Glyzin, “Razmernostnye kharakteristiki diffuzionnogo khaosa”, Model. i analiz inform. sistem, 20:1 (2013), 30–51
Kulikov A.N., “Landau-Hopf scenario of passage to turbulence in some problems of elastic stability theory”, Differ. Equ., 48:9 (2012), 1258–1271
A. S. Bobok, S. D. Glyzin, “Avtokolebaniya reshetok nelineinykh elementov v opyte Skotta”, Model. i analiz inform. sistem, 19:5 (2012), 56–68
A. P. Kuznetsov, S. P. Kuznetsov, L. V. Tyuryukina, I. R. Sataev, “Stsenarii Landau–Khopfa v ansamble vzaimodeistvuyuschikh ostsillyatorov”, Nelineinaya dinam., 8:5 (2012), 863–873
S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “Finite-dimensional models of diffusion chaos”, Comput. Math. Math. Phys., 50:5 (2010), 816–830
A. Yu. Kolesov, N. Kh. Rozov, “On the definition of ‘chaos’”, Russian Math. Surveys, 64:4 (2009), 701–744
E. S. Kokuikin, A. N. Kulikov, “Tsikly i tory delovoi aktivnosti v odnoi matematicheskoi modeli makroekonomiki”, Model. i analiz inform. sistem, 16:4 (2009), 86–95
S. M. Aldoshin, D. V. Anosov, G. G. Chernyi, V. N. Chubarikov, S. V. Emel'yanov, L. D. Faddeev, A. T. Fomenko, A. A. Gonchar, A. I. Grigor'ev, V. A. Il'in, M. P. Kirpichnikov, S. K. Korovin, M. V. Koval'chuk, V. V. Kozlov, A. B. Kurzhanskii, G. I. Marchuk, V. P. Maslov, E. I. Moiseev, S. M. Nikol'skii, S. P. Novikov, Yu. M. Okunev, Yu. S. Osipov, B. E. Paton, V. P. Skulachev, V. A. Tkachuk, E. P. Velikhov, Yu. I. Zhuravlev, “Viktor Antonovich Sadovnichii. A tribute in honor of his seventieth birthday”, Diff Equat, 45:4 (2009), 465
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