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.
\Bibitem{KolRoz09}
\by A.~Yu.~Kolesov, N.~Kh.~Rozov
\paper On the definition of `chaos'
\jour Russian Math. Surveys
\yr 2009
\vol 64
\issue 4
\pages 701--744
\mathnet{http://mi.mathnet.ru/eng/rm9297}
\crossref{https://doi.org/10.1070/RM2009v064n04ABEH004631}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2583574}
\zmath{https://zbmath.org/?q=an:05665290}
\adsnasa{https://adsabs.harvard.edu/cgi-bin/bib_query?2009RuMaS..64..701K}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000275492400004}
\elib{https://elibrary.ru/item.asp?id=20425307}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-77951191412}
Linking options:
https://www.mathnet.ru/eng/rm9297
https://doi.org/10.1070/RM2009v064n04ABEH004631
https://www.mathnet.ru/eng/rm/v64/i4/p125
This publication is cited in the following 19 articles:
Nina I. Zhukova, “Sensitivity and Chaoticity of Some Classes of Semigroup Actions”, Regul. Chaotic Dyn., 29:1 (2024), 174–189
Tom Eivind Glover, Ruben Jahren, Francesco Martinuzzi, Pedro Gonçalves Lind, Stefano Nichele, “A sensitivity analysis of cellular automata and heterogeneous topology networks: partially-local cellular automata and homogeneous homogeneous random boolean networks”, International Journal of Parallel, Emergent and Distributed Systems, 2024, 1
Vladimir Anashin, “Free Choice in Quantum Theory: A p-adic View”, Entropy, 25:5 (2023), 830
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
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
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
Hirsch M.W., “on the Nonchaotic Nature of Monotone Dynamical Systems”, Eur. J. Pure Appl Math., 12:3 (2019), 680–688
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
S. D. Glyzin, “Dimensional Characteristics of Diffusion Chaos”, Model. anal. inf. sist., 20:1 (2015), 30
Christian Kuehn, Applied Mathematical Sciences, 191, Multiple Time Scale Dynamics, 2015, 431
V. E. Kim, “Dynamics of linear operators connected with su(1,1) algebra”, Ufa Math. J., 6:1 (2014), 66–70
S. D. Glyzin, “Razmernostnye kharakteristiki diffuzionnogo khaosa”, Model. i analiz inform. sistem, 20:1 (2013), 30–51
S. D. Glyzin, “Dimensional characteristics of diffusion chaos”, Aut. Control Comp. Sci., 47:7 (2013), 452–469
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
Schneider F.M., Kerkhoff S., Behrisch M., Siegmund S., “Chaotic Actions of Topological Semigroups”, Semigr. Forum, 87:3 (2013), 590–598
Stewart I., “Sources of uncertainty in deterministic dynamics: an informal overview”, Phil. Trans. R. Soc. A, 369:1956 (2011), 4705–4729
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
A. Yu. Loskutov, “Fascination of chaos”, Phys. Usp., 53:12 (2010), 1257–1280
S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “Finite-dimensional models of diffusion chaos”, Comput. Math. Math. Phys., 50:5 (2010), 816–830
Citation in format AMSBIB
Contact us: