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Russian Mathematical Surveys, 2009, Volume 64, Issue 4, Pages 701–744
DOI: https://doi.org/10.1070/RM2009v064n04ABEH004631 (Mi rm9297) 		 
This article is cited in 19 scientific papers (total in 19 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
English version PDF (733 kB) Citations (19)
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DOI: https://doi.org/10.1070/RM2009v064n04ABEH004631
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.
Received: 07.05.2009
Russian version:
Uspekhi Matematicheskikh Nauk, 2009, Volume 64, Issue 4(388), Pages 125–172
DOI: https://doi.org/10.4213/rm9297
Bibliographic databases:
Document Type: Article
UDC: 517.957
MSC: Primary 37D45; Secondary 37A25, 34D45, 37A35, 37D10
Language: English
Original paper language: Russian
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
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    • This publication is cited in the following 19 articles:
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