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Uspekhi Mat. Nauk, 2009, Volume 64, Issue 4(388), Pages 125–172 (Mi umn9297)  

This article is cited in 14 scientific papers (total in 14 papers)

On the definition of ‘chaos’

A. Yu. Kolesova, N. Kh. Rozovb

a P. G. Demidov Yaroslavl State University
b M. V. Lomonosov Moscow State University

Abstract: A new definition of a chaotic invariant set is given for a continuous semiflow in a metric space. It generalizes the well-known definition due to Devaney and allows one to take into account a special feature occurring in the non-compact infinite-dimensional case: so-called turbulent chaos. The paper consists of two sections. The first contains several well-known facts from chaotic dynamics, together with new definitions and results. The second presents a concrete example demonstrating that our definition of chaos is meaningful. Namely, an infinite-dimensional system of ordinary differential equations is investigated having an attractor that is chaotic in the sense of the new definition but not in the sense of Devaney or Knudsen.
Bibliography: 65 titles.

Keywords: attractor, chaos, topological transitivity, mixing, invariant measure, hyperbolicity.


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English version:
Russian Mathematical Surveys, 2009, 64:4, 701–744

Bibliographic databases:

UDC: 517.957
MSC: Primary 37D45; Secondary 37A25, 34D45, 37A35, 37D10
Received: 07.05.2009

Citation: A. Yu. Kolesov, N. Kh. Rozov, “On the definition of ‘chaos’”, Uspekhi Mat. Nauk, 64:4(388) (2009), 125–172; Russian Math. Surveys, 64:4 (2009), 701–744

Citation in format AMSBIB
\by A.~Yu.~Kolesov, N.~Kh.~Rozov
\paper On the definition of `chaos'
\jour Uspekhi Mat. Nauk
\yr 2009
\vol 64
\issue 4(388)
\pages 125--172
\jour Russian Math. Surveys
\yr 2009
\vol 64
\issue 4
\pages 701--744

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    1. A. Yu. Perevaryukha, “Perekhod k ustoichivomu khaoticheskomu rezhimu v novoi modeli dinamiki populyatsii v rezultate edinstvennoi bifurkatsii”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2010, no. 2, 117–126  mathnet
    2. A. Yu. Loskutov, “Fascination of chaos”, Phys. Usp., 53:12 (2010), 1257–1280  mathnet  crossref  crossref  isi  elib
    3. S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “Finite-dimensional models of diffusion chaos”, Comput. Math. Math. Phys., 50:5 (2010), 816–830  mathnet  crossref  adsnasa  isi  elib  elib
    4. Stewart I., “Sources of uncertainty in deterministic dynamics: an informal overview”, Phil. Trans. R. Soc. A, 369:1956 (2011), 4705–4729  crossref  mathscinet  zmath  adsnasa  isi  scopus
    5. S. D. Glyzin, “Razmernostnye kharakteristiki diffuzionnogo khaosa”, Model. i analiz inform. sistem, 20:1 (2013), 30–51  mathnet
    6. S. D. Glyzin, “Dimensional characteristics of diffusion chaos”, Aut. Control Comp. Sci., 47:7 (2013), 452–469  crossref  mathscinet  scopus
    7. Schneider F.M., Kerkhoff S., Behrisch M., Siegmund S., “Locally Compact Groups Admitting Faithful Strongly Chaotic Actions on Hausdorff Spaces”, Int. J. Bifurcation Chaos, 23:9 (2013), 1350158  crossref  mathscinet  zmath  isi  scopus
    8. Schneider F.M., Kerkhoff S., Behrisch M., Siegmund S., “Chaotic Actions of Topological Semigroups”, Semigr. Forum, 87:3 (2013), 590–598  crossref  mathscinet  zmath  isi  scopus
    9. V. E. Kim, “Dynamics of linear operators connected with $\mathrm{su}(1,1)$ algebra”, Ufa Math. J., 6:1 (2014), 66–70  mathnet  crossref  elib
    10. Zheng J., Hu H., Xia X., “Applications of Symbolic Dynamics in Counteracting the Dynamical Degradation of Digital Chaos”, Nonlinear Dyn., 94:2 (2018), 1535–1546  crossref  isi  scopus
    11. Hirsch M.W., “on the Nonchaotic Nature of Monotone Dynamical Systems”, Eur. J. Pure Appl Math., 12:3 (2019), 680–688  crossref  isi
    12. S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “Diffusion chaos and its invariant numerical characteristics”, Theoret. and Math. Phys., 203:1 (2020), 443–456  mathnet  crossref  crossref  isi  elib
    13. A. S. Sheludko, “Convergence analysis of the guaranteed parameter estimation algorithm for models of one-dimensional chaotic systems”, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 13:2 (2020), 144–150  mathnet  crossref
    14. Zheng J., Hu H., Ming H., Liu X., “Theoretical Design and Circuit Implementation of Novel Digital Chaotic Systems Via Hybrid Control”, Chaos Solitons Fractals, 138 (2020), 109863  crossref  mathscinet  isi
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