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Nelin. Dinam., 2009, Volume 5, Number 3, Pages 403–424 (Mi nd101)  

This article is cited in 8 scientific papers (total in 8 papers)

An example of a non-autonomous continuous-time system with attractor of Plykin type in the Poincaré map

S. P. Kuznetsov

Saratov Branch of Institute of Radio-engineering and Electronics, Russian Academy of Sciences

Abstract: A non-autonomous flow system is introduced, which may serve as a base for elaboration of real systems and devices demonstrating the structurally stable chaotic dynamics. The starting point is the map of the sphere composed of four stages of sequential continuous geometrically evident transformations. The computations indicate that in a certain parameter range the map posseses an attractor of Plykin type. Accounting the structural stability intrinsic to this attractor, a modification of the model is undertaken, which includes a variable change with passage to representation of instantaneous states on the plane. As a result, a set of two non-autonomous differential equations of the first order with smooth coefficients is obtained explicitly, which has the Plykin type attractor in the plane in the Poincaré cross-section. Results of computations are presented for the sphere map and for the flow system including portraits of attractors, Lyapunov exponents, dimension estimates. Substantiation of the hyperbolic nature of the attractors for the sphere map and for the flow system is based on a computer procedure of verification of the so-called cone criterion; in this context, some hints are applied, which may be useful in similar computations for some other systems.

Keywords: hyperbolic chaos, Plykin attractor, Lyapunov exponent, structural stability.

Full text: PDF file (2310 kB)
UDC: 517.9
MSC: 37D45
Received: 14.10.2008

Citation: S. P. Kuznetsov, “An example of a non-autonomous continuous-time system with attractor of Plykin type in the Poincaré map”, Nelin. Dinam., 5:3 (2009), 403–424

Citation in format AMSBIB
\by S.~P.~Kuznetsov
\paper An example of a non-autonomous continuous-time system with attractor of Plykin type in the Poincar{\'e} map
\jour Nelin. Dinam.
\yr 2009
\vol 5
\issue 3
\pages 403--424

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    This publication is cited in the following articles:
    1. Kuznetsov S.P., “Giperbolicheskie strannye attraktory sistem dopuskayuschikh fizicheskuyu realizatsiyu”, Izv. vuzov. Prikladnaya nelineinaya dinamika, 17:4 (2009), 5–34  zmath  elib
    2. A. Yu. Loskutov, “Fascination of chaos”, Phys. Usp., 53:12 (2010), 1257–1280  mathnet  crossref  crossref  isi  elib
    3. Kuznetsov A.S. Kuznetsov S.P., Sataev I.R., “Parametric generator of hyperbolic chaos based on two coupled oscillators with nonlinear dissipation”, Technical Physics, 55:12 (2010), 1707–1715  crossref  adsnasa  isi  elib  scopus
    4. S. P. Kuznetsov, “Dynamical chaos and uniformly hyperbolic attractors: from mathematics to physics”, Phys. Usp., 54:2 (2011), 119–144  mathnet  crossref  crossref  adsnasa  isi  elib
    5. Kuznetsov S.P., “Plykin type attractor in electronic device simulated in MULTISIM”, Chaos, 21:4 (2011), 043105, 8 pp.  crossref  zmath  adsnasa  isi  elib  scopus
    6. Arzhanukhina D.S., “O stsenariyakh razrusheniya giperbolicheskogo khaosa v modelnykh otobrazheniyakh na tore s dissipativnym vozmuscheniem”, Izvestiya vysshikh uchebnykh zavedenii. Prikladnaya nelineinaya dinamika, 20:1 (2012), 117–123  zmath  elib
    7. Arzhanukhina D.S., Kuznetsov S.P., “Sistema trekh neavtonomnykh ostsillyatorov s giperbolicheskim khaosom. chast i. model s dinamikoi na attraktore, opisyvaemoi otobrazheniem na tore kot arnolda”, Izvestiya vysshikh uchebnykh zavedenii. prikladnaya nelineinaya dinamika, 20:6 (2012), 56–66  zmath  elib
    8. A. Yu. Zhalnin, “Ot kvazigarmonicheskikh ostsillyatsii k neironnym spaikam i berstam: raznoobrazie rezhimov giperbolicheskogo khaosa na osnove attraktora Smeila Vilyamsa”, Nelineinaya dinam., 12:1 (2016), 53–73  mathnet
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