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 Regul. Chaotic Dyn., 2015, Volume 20, Issue 6, Pages 649–666 (Mi rcd35)

Hyperbolic Chaos in Self-oscillating Systems Based on Mechanical Triple Linkage: Testing Absence of Tangencies of Stable and Unstable Manifolds for Phase Trajectories

Sergey P. Kuznetsovab

a Udmurt State University, ul. Universitetskaya 1, Izhevsk, 426034 Russia
b Kotelnikov’s Institute of Radio-Engineering and Electronics of RAS, Saratov Branch, ul. Zelenaya 38, Saratov, 410019 Russia

Abstract: Dynamical equations are formulated and a numerical study is provided for selfoscillatory model systems based on the triple linkage hinge mechanism of Thurston – Weeks – Hunt – MacKay. We consider systems with a holonomic mechanical constraint of three rotators as well as systems, where three rotators interact by potential forces. We present and discuss some quantitative characteristics of the chaotic regimes (Lyapunov exponents, power spectrum). Chaotic dynamics of the models we consider are associated with hyperbolic attractors, at least, at relatively small supercriticality of the self-oscillating modes; that follows from numerical analysis of the distribution for angles of intersection of stable and unstable manifolds of phase trajectories on the attractors. In systems based on rotators with interacting potential the hyperbolicity is violated starting from a certain level of excitation.

Keywords: dynamical system, chaos, hyperbolic attractor, Anosov dynamics, rotator, Lyapunov exponent, self-oscillator

 Funding Agency Grant Number Russian Science Foundation 15-12-20035 This work was supported by RSF grant No 15-12-20035.

DOI: https://doi.org/10.1134/S1560354715060027

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Bibliographic databases:

MSC: 37D45, 37D20, 34D08, 32Q05, 70F20
Accepted:30.10.2015
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Citation: Sergey P. Kuznetsov, “Hyperbolic Chaos in Self-oscillating Systems Based on Mechanical Triple Linkage: Testing Absence of Tangencies of Stable and Unstable Manifolds for Phase Trajectories”, Regul. Chaotic Dyn., 20:6 (2015), 649–666

Citation in format AMSBIB
\Bibitem{Kuz15} \by Sergey P. Kuznetsov \paper Hyperbolic Chaos in Self-oscillating Systems Based on Mechanical Triple Linkage: Testing Absence of Tangencies of Stable and Unstable Manifolds for Phase Trajectories \jour Regul. Chaotic Dyn. \yr 2015 \vol 20 \issue 6 \pages 649--666 \mathnet{http://mi.mathnet.ru/rcd35} \crossref{https://doi.org/10.1134/S1560354715060027} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3431181} \adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2015RCD....20..649K} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000365809000002} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84948967074} 

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• http://mi.mathnet.ru/eng/rcd/v20/i6/p649

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles
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This publication is cited in the following articles:
1. Sergey P. Kuznetsov, Vyacheslav P. Kruglov, “Verification of Hyperbolicity for Attractors of Some Mechanical Systems with Chaotic Dynamics”, Regul. Chaotic Dyn., 21:2 (2016), 160–174
2. S. P. Kuznetsov, “From geodesic flow on a surface of negative curvature to electronic generator of robust chaos”, Int. J. Bifurcation Chaos, 26:14 (2016), 1650232
3. P. V. Kuptsov, S. P. Kuznetsov, “Numerical test for hyperbolicity of chaotic dynamics in time-delay systems”, Phys. Rev. E, 94:1 (2016), 010201
4. S. P. Kuznetsov, V. P. Kruglov, “On some simple examples of mechanical systems with hyperbolic chaos”, Proc. Steklov Inst. Math., 297 (2017), 208–234
5. P. V. Kuptsov, S. P. Kuznetsov, “Numerical test for hyperbolicity in chaotic systems with multiple time delays”, Commun. Nonlinear Sci. Numer. Simul., 56 (2018), 227–239