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 Regul. Chaotic Dyn., 2018, Volume 23, Issue 7-8, Pages 908–932 (Mi rcd374)

Lyapunov Analysis of Strange Pseudohyperbolic Attractors: Angles Between Tangent Subspaces, Local Volume Expansion and Contraction

Pavel V. Kuptsova, Sergey P. Kuznetsovbc

a Institute of electronics and mechanical engineering, Yuri Gagarin State Technical University of Saratov ul. Politekhnicheskaya 77, Saratov, 410054 Russia
b Kotel’nikov’s Institute of Radio-Engineering and Electronics of RAS, Saratov Branch, ul. Zelenaya 38, Saratov, 410019 Russia
c Udmurt State University, ul. Universitetskaya 1, Izhevsk, 426034 Russia

Abstract: Pseudohyperbolic attractors are genuine strange chaotic attractors. They do not contain stable periodic orbits and are robust in the sense that such orbits do not appear under variations. The tangent space of these attractors is split into a direct sum of volume expanding and contracting subspaces and these subspaces never have tangencies with each other. Any contraction in the first subspace, if it occurs, is weaker than contractions in the second one. In this paper we analyze the local structure of several chaotic attractors recently suggested in the literature as pseudohyperbolic. The absence of tangencies and thus the presence of the pseudohyperbolicity is verified using the method of angles that includes computation of distributions of the angles between the corresponding tangent subspaces. Also, we analyze how volume expansion in the first subspace and the contraction in the second one occurs locally. For this purpose we introduce a family of instant Lyapunov exponents. Unlike the well-known finite time ones, the instant Lyapunov exponents show expansion or contraction on infinitesimal time intervals. Two types of instant Lyapunov exponents are defined. One is related to ordinary finite-time Lyapunov exponents computed in the course of standard algorithm for Lyapunov exponents. Their sums reveal instant volume expanding properties. The second type of instant Lyapunov exponents shows how covariant Lyapunov vectors grow or decay on infinitesimal time. Using both instant and finite-time Lyapunov exponents, we demonstrate that average expanding and contracting properties specific to pseudohyperbolicity are typically violated on infinitesimal time. Instantly volumes from the first subspace can sometimes be contacted, directions in the second subspace can sometimes be expanded, and the instant contraction in the first subspace can sometimes be stronger than the contraction in the second subspace.

Keywords: chaotic attractor, strange pseudohyperbolic attractor, method of angles, hyperbolic isolation, Lyapunov exponents, finite-time Lyapunov exponents, instant Lyapunov exponents, covariant Lyapunov vectors

 Funding Agency Grant Number Russian Science Foundation 15-12-20035 Russian Foundation for Basic Research 16-02-00135 The work of SPK on theoretical formulations was supported by the Russian Science Foundation under grant No 15-12-20035. The work of PVK on elaborating computer routines and numerical computations was supported by RFBR under grant No 16-02-00135.

DOI: https://doi.org/10.1134/S1560354718070079

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MSC: 37D45,37D30,37D25,65L99,34D08
Accepted:06.11.2018
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Citation: Pavel V. Kuptsov, Sergey P. Kuznetsov, “Lyapunov Analysis of Strange Pseudohyperbolic Attractors: Angles Between Tangent Subspaces, Local Volume Expansion and Contraction”, Regul. Chaotic Dyn., 23:7-8 (2018), 908–932

Citation in format AMSBIB
\Bibitem{KupKuz18} \by Pavel V. Kuptsov, Sergey P. Kuznetsov \paper Lyapunov Analysis of Strange Pseudohyperbolic Attractors: Angles Between Tangent Subspaces, Local Volume Expansion and Contraction \jour Regul. Chaotic Dyn. \yr 2018 \vol 23 \issue 7-8 \pages 908--932 \mathnet{http://mi.mathnet.ru/rcd374} \crossref{https://doi.org/10.1134/S1560354718070079} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000458183900007} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85061267556} 

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