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On Discrete Homoclinic Attractors of Three-Dimensional Diffeomorphisms

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Abstract

We give a short review on discrete homoclinic attractors. Such strange attractors contain only one saddle fixed point and, hence, entirely its unstable invariant manifold. We discuss the most important peculiarities of these attractors such as their geometric and homoclinic structures, phenomenological scenarios of their appearance, pseudohyperbolic properties etc.

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Notes

  1. Recall that in the case of the classical Lorenz attractor, a (trivial) lacuna is an open region (a hole in the attractor) that contains a saddle limit cycle whose the stable manifold does not intersect the attractor [16]. Accordingly, in the case of discrete Lorenz attractor, a lacuna contains a saddle closed invariant curve with the same property. When varying parameters, lacunae in attractor can appear and disappear due to the appearance/disappearance of homoclinic intersections. The corresponding bifurcations belong to the class of the so-called internal bifurcations of attractor. See more details in [16–18].

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Funding

This paper was carried out in the framework of the grant 19-11-00280 of the RSciF. Section 3.3.1 was supported by the grant 20-71-00079 of the RSciF and Section 2.2.1 by the RFBR grant 18-29-00081. The authors thank the Russian Ministry of Science and Education (grant 0729-2020-0036) for supporting scientific researches. S. Gonchenko thanks the Theoretical Physics and Mathematics Advancement Foundation ‘‘BASIS’’ for support of scientific investigations of his group.

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Authors and Affiliations

  1. Lobachevsky State University of Nizhny Novgorod, 603022, Nizhny Novgorod, Russia

    A. S. Gonchenko & S. V. Gonchenko

  2. National Research University Higher School of Economics, 603155, Nizhny Novgorod, Russia

    S. V. Gonchenko

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  1. A. S. Gonchenko
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  2. S. V. Gonchenko
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Correspondence to A. S. Gonchenko or S. V. Gonchenko.

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(Submitted by A. B. Muravnik)

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Gonchenko, A.S., Gonchenko, S.V. On Discrete Homoclinic Attractors of Three-Dimensional Diffeomorphisms. Lobachevskii J Math 42, 3352–3364 (2021). https://doi.org/10.1134/S1995080222020068

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