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On the Birth of Discrete Lorenz Attractors Under Bifurcations of 3D Maps with Nontransversal Heteroclinic Cycles

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Abstract

Lorenz attractors are important objects in the modern theory of chaos. The reason, on the one hand, is that they are encountered in various natural applications (fluid dynamics, mechanics, laser dynamics, etc.). On the other hand, Lorenz attractors are robust in the sense that they are generally not destroyed by small perturbations (autonomous, nonautonomous, stochastic). This allows us to be sure that the object observed in the experiment is exactly a chaotic attractor rather than a long-time periodic orbit.

Discrete-time analogs of the Lorenz attractor possess even more complicated structure — they allow homoclinic tangencies of invariant manifolds within the attractor. Thus, discrete Lorenz attractors belong to the class of wild chaotic attractors. These attractors can be born in codimension-three local and certain global (homoclinic and heteroclinic) bifurcations. While various homoclinic bifurcations leading to such attractors have been studied, for heteroclinic cycles only cases where at least one of the fixed points is a saddle-focus have been considered to date.

In the present paper the case of a heteroclinic cycle consisting of saddle fixed points with a quadratic tangency of invariant manifolds is considered. It is shown that, in order to have three-dimensional chaos such as the discrete Lorenz attractors, one needs to avoid the existence of lower-dimensional global invariant manifolds. Thus, it is assumed that either the quadratic tangency or the transversal heteroclinic orbit is nonsimple. The main result of the paper is the proof that the original system is the limiting point in the space of dynamical systems of a sequence of domains in which the diffeomorphism possesses discrete Lorenz attractors.

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Notes

  1. Note that in [4] a geometric model of the four-dimensional system with a wild spiral attractor was constructed. However, specific examples of systems of differential equations with such attractors had not been known for a long time. Quite recently, an example of such systems was given in [5] — a four-dimensional extension of the Lorenz model.

  2. The detailed definition of simple and nonsimple heteroclinic orbits will be given below in this section.

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ACKNOWLEDGMENTS

The author thanks S. V. Gonchenko for posing the problem considered in this paper. Also, the author thanks J. C. Tatjer for useful discussions that led to the idea of condition D2 and Case III, and two anonymous reviewers, whose comments helped to improve the paper.

Funding

This paper is a contribution to the project M7 (Dynamics of Geophysical Problems in Turbulent Regimes) of the Collaborative Research Center TRR 181 “Energy Transfer in Atmosphere and Ocean” funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) — project number 274762653. This work was also supported by the grant of the Russian Science Foundation 19-11-00280.

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  1. University of Bremen, MARUM, Department of Mathematics, Bibliothekstrasse 5, 28359, Bremen, Germany

    Ivan I. Ovsyannikov

  2. Lobachevsky State University of Nizhny Novgorod, ITMM, pr. Gagarina 23, 603022, Nizhny Novgorod, Russia

    Ivan I. Ovsyannikov

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Correspondence to Ivan I. Ovsyannikov.

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MSC2010

37C05, 37G25, 37G35

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Ovsyannikov, I.I. On the Birth of Discrete Lorenz Attractors Under Bifurcations of 3D Maps with Nontransversal Heteroclinic Cycles. Regul. Chaot. Dyn. 27, 217–231 (2022). https://doi.org/10.1134/S156035472202006X

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