Abstract:
The topic of the paper is strange attractors of multidimensional maps and flows. Strange attractors can be divided into two groups: genuine attractors, that keep their chaoticity under small perturbations, and quasi-attractors (according to Afraimovich-Shilnikov), inside which stable periodic orbits can arise under small perturbations. Main goal of this work is to construct effective criteria that make it possible to distinguish such attractors, as well as to verify these criteria by means of numerical experiments. Under "genuine" attractors, we mean the so-called pseudohyperbolic attractors. We give their definition and describe characteristic properties, on the basis of which two numerical methods are constructed, which allow to check the principally important property of pseudohyperbolic attractors: the continuity of strong contracting subspaces and subspaces where volumes are expanded. As examples on which numerical methods for checking pseudohyperbolicity have been tested, we consider the classical Henon map, the singularly hyperbolic Lozi map, the Anosov diffeomorphism of two-dimensional torus, the classical Lorenz and Shimizu-Morioka systems, as well as a three-dimensional Henon-like maps.
This work was supported by the RSF grants No. 19-11-00280 (Introduction, Sections 1 and 3.3)
and 19-71-10048 (Sections 2 and 3). Numerical results with models in Section 3 were supported by the Laboratory
of Dynamical Systems and Applications NRU HSE, of the Russian Ministry of Science and Higher Education (Grant
No. 075-15-2019-1931). Authors also thank RFBR (grants 18-29-10081 and 19-01-00607) and the Theoretical Physics and
Mathematics Advancement Foundation «BASIS». The authors also thank P.V. Kuptsov for fruitful discussion and useful
comments.
Received: 18.12.2020
Bibliographic databases:
Document Type:
Article
UDC:517.925 + 517.93
Language: Russian
Citation:
S. V. Gonchenko, M. H. Kaynov, A. O. Kazakov, D. V. Turaev, “On methods for verification of the pseudohyperbolicity of strange attractors”, Izvestiya VUZ. Applied Nonlinear Dynamics, 29:1 (2021), 160–185
\Bibitem{GonKayKaz21}
\by S.~V.~Gonchenko, M.~H.~Kaynov, A.~O.~Kazakov, D.~V.~Turaev
\paper On methods for verification of the pseudohyperbolicity of strange attractors
\jour Izvestiya VUZ. Applied Nonlinear Dynamics
\yr 2021
\vol 29
\issue 1
\pages 160--185
\mathnet{http://mi.mathnet.ru/ivp406}
\crossref{https://doi.org/10.18500/0869-6632-2021-29-1-160-185}
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This publication is cited in the following 7 articles:
S. D. Glyzin, A. Yu. Kolesov, “Symmetric hyperbolic trap”, Math. Notes, 116:3 (2024), 446–457
Alexey Kazakov, Dmitrii Mints, Iuliia Petrova, Oleg Shilov, “On non-trivial hyperbolic sets and their bifurcations in families of diffeomorphisms of a two-dimensional torus”, Chaos: An Interdisciplinary Journal of Nonlinear Science, 34:8 (2024)
Russian Math. Surveys, 78:4 (2023), 635–777
M. V. Demina, D. O. Ilyukhin, “Invariantnye algebraicheskie mnogoobraziya dlya modeli dvoinoi konvektsii Raklidzha”, Sib. matem. zhurn., 64:5 (2023), 982–991
Sergey Gonchenko, Efrosiniia Karatetskaia, Alexey Kazakov, Vyacheslav Kruglov, “Conjoined Lorenz twins—a new pseudohyperbolic attractor in three-dimensional maps and flows”, Chaos: An Interdisciplinary Journal of Nonlinear Science, 32:12 (2022)
S. D. Glyzin, A. Yu. Kolesov, “On some modifications of Arnold's cat map”, Dokl. Math., 104:2 (2021), 242–246
Alexey Kazakov, “On bifurcations of Lorenz attractors in the Lyubimov–Zaks model”, Chaos: An Interdisciplinary Journal of Nonlinear Science, 31:9 (2021)
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