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MSC2010 34C23, 34D45, 65P20

The asymmetric Lorenz attractor as an example of a new pseudohyperbolic attractor of three-dimensional systems

A. O. Kazakov A. D. Kozlov1

AnnotationIn the paper a new method of constructing of three-dimensional flow systems with different chaotic attractors is presented. Using this method, an example of three-dimensional system possessing an asymmetric Lorenz attractor is obtained. Unlike the classical Lorenz attractor, the observed attractor does not have symmetry. However, the discovered asymmetric attractor, as well as classical one, belongs to a class of <<true>> chaotic, or, more precise, pseudohyperbolic attractors; the theory of such attractors was developed by D.~Turaev and L.P.~Shilnikov. Any trajectory of a pseudohyperbolic attractor has a positive Lyapunov exponent and this property holds for attractors of close systems. In this case, in contrast to hyperbolic attractors, pseudohyperbolic ones admit homoclinic tangencies, but bifurcations of such tangencies do not lead to generation of stable periodic orbits. In order to find the non-symmetric Lorenz attractor we applied the method of <<saddle chart>>. Using diagrams of maximal Lyapunov exponent, we show that there are no stability windows in the neighborhood of the observed attractor. In addition, we verify the pseudohyperbolicity for the non-symmetric Lorenz attractor using the LMP-method developed quite recently by Gonchenko, Kazakov and Turaev.
Keywordsstrange attractor, pseudohyperbolicity, homoclinic orbit, Lorenz attractor, Lyapunov exponents

1Alexey O. Kazakov, Senior Researcher, Laboratory of Topological Methods in Dynamics, National Research University Higher School of Economics (25/12 Bolshaya Pecherskaja St., Nizhny Novgorod 603155, Russia); Assistant Prosfessor, Nanional Research Lobachevsky State University of Nizhny Novgorod (23 Prospekt Gagarina 603950 Nizhny Novgorod, Russia); PhD, ORCID: 0000-0003-0002-6553, kazakovdz@yandex.ru

2Alexander D. Kozlov, Junior Researcher, Laboratory of Topological methods in Dynamics, National Research University Higher School of Economics (25/12 Bolshaya Pecherskaya St., Nizhny Novgorod 603155, Russia); PhD Student, (23 Prospekt Gagarina 603950 Nizhny Novgorod, Russia); ORCID: 0000-0003-1830-4769, kozzzloff@list.ru

Citation: A. O. Kazakov A. D. Kozlov, "[The asymmetric Lorenz attractor as an example of a new pseudohyperbolic attractor of three-dimensional systems]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,20:2 (2018) 187–198 (In Russian)

DOI 10.15507/2079-6900.20.201802.187-198