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Gonchenko S. V., Kainov M. N., Kazakov A. O., Turaev D. V. On methods for verification of the pseudohyperbolicity of strange attractors. Izvestiya VUZ. Applied Nonlinear Dynamics, 2021, vol. 29, iss. 1, pp. 160-185. DOI: 10.18500/0869-6632-2021-29-1-160-185

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On methods for verification of the pseudohyperbolicity of strange attractors

Gonchenko Sergey Vladimirovich, Lobachevsky State University of Nizhny Novgorod
Kainov M. N., National Research University "Higher School of Economics"
Kazakov Aleksej Olegovich, National Research University "Higher School of Economics"
Turaev D. V., Imperial College London

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.
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