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UFN, 2011, Volume 181, Number 2, Pages 121–149 (Mi ufn2331)  

This article is cited in 43 scientific papers (total in 43 papers)


Dynamical chaos and uniformly hyperbolic attractors: from mathematics to physics

S. P. Kuznetsov

Saratov Branch, Kotel'nikov Institute of Radio-Engineering and Electronics, Russian Academy of Sciences

Abstract: Research is reviewed on the identification and construction of physical systems with chaotic dynamics due to uniformly hyperbolic attractors (such as the Plykin attraction or the Smale–Williams solenoid). Basic concepts of the mathematics involved and approaches proposed in the literature for constructing systems with hyperbolic attractors are discussed. Topics covered include periodic pulse-driven models; dynamics models consisting of periodically repeated stages, each described by its own differential equations; the construction of systems of alternately excited coupled oscillators; the use of parametrically excited oscillations; and the introduction of delayed feedback. Some maps, differential equations, and simple mechanical and electronic systems exhibiting chaotic dynamics due to the presence of uniformly hyperbolic attractors are presented as examples.


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Physics–Uspekhi, 2011, 54:2, 119–144

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PACS: 05.45.-a, 45.50.-j, 84.30.-r
Received: April 1, 2010
Revised: September 16, 2010

Citation: S. P. Kuznetsov, “Dynamical chaos and uniformly hyperbolic attractors: from mathematics to physics”, UFN, 181:2 (2011), 121–149; Phys. Usp., 54:2 (2011), 119–144

Citation in format AMSBIB
\by S.~P.~Kuznetsov
\paper Dynamical chaos and uniformly hyperbolic attractors: from mathematics to physics
\jour UFN
\yr 2011
\vol 181
\issue 2
\pages 121--149
\jour Phys. Usp.
\yr 2011
\vol 54
\issue 2
\pages 119--144

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    This publication is cited in the following articles:
    1. Olga Isaeva, Sergey Kuznetsov, Erik Mosekilde, “Hyperbolic chaotic attractor in amplitude dynamics of coupled self-oscillators with periodic parameter modulation”, Phys. Rev. E, 84:1 (2011), 016228, 10 pp.  crossref  isi  scopus
    2. Sergey P. Kuznetsov, “Plykin type attractor in electronic device simulated in MULTISIM”, Chaos, 21:4 (2011), 043105  crossref  zmath  isi  scopus
    3. Kuznetsov S.P., “Skhemy elektronnykh ustroistv s giperbolicheskim khaosom i modelirovanie ikh dinamiki v programmnoi srede multisim”, Izvestiya vysshikh uchebnykh zavedenii. Prikladnaya nelineinaya dinamika, 19:5 (2011), 98–115  zmath  elib
    4. A. Yu. Loskutov, A. V. Popkova, “Stabilization of chaotic oscillations in systems with a hyperbolic-type attractor”, JETP Letters, 94:1 (2011), 86–90  mathnet  crossref  isi
    5. P. Kuptsov, “Fast numerical test of hyperbolic chaos”, Phys. Rev. E, 85:1 (2012), 015203(R), 4 pp.  crossref  adsnasa  isi  elib  scopus
    6. P. Kuptsov, S. Kuznetsov, A. Pikovsky, “Hyperbolic Chaos of Turing Patterns”, Phys. Rev. Lett, 108:19 (2012), 194101, 4 pp.  crossref  adsnasa  isi  elib  scopus
    7. A.V. Makarenko, “Measure of synchronism of multidimensional chaotic sequences based on their symbolic representation in a T-alphabet”, Tech. Phys. Lett, 38:9 (2012), 804  crossref  adsnasa  isi  elib  scopus
    8. O.B. Isaeva, S.P. Kuznetsov, I.R. Sataev, “A “saddle-node” bifurcation scenario for birth or destruction of a Smale–Williams solenoid”, Chaos, 22:4 (2012), 043111, 7 pp.  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    9. 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
    10. Kruglov V.P., “Attraktor tipa smeila-vilyamsa v koltsevoi sisteme s periodicheskoi modulyatsiei chastoty”, Izvestiya vysshikh uchebnykh zavedenii. prikladnaya nelineinaya dinamika, 20:1 (2012), 124–128  mathscinet  zmath  elib
    11. Kuznetsov A.S., “Parametricheskie generatory s khaoticheskoi amplitudnoi dinamikoi, otvechayuschei attraktoram tipa smeila-vilyamsa”, Izvestiya vysshikh uchebnykh zavedenii. prikladnaya nelineinaya dinamika, 20:1 (2012), 129–136  mathscinet  zmath  elib
    12. 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
    13. A.S. Kuznetsov, S.P. Kuznetsov, “Parametric generation of robust chaos with time-delayed feedback and modulated pump source”, Communications in Nonlinear Science and Numerical Simulation, 18:3 (2013), 728–734  crossref  mathscinet  zmath  adsnasa  isi  scopus
    14. Pavel V. Kuptsov, Sergey P. Kuznetsov, Arkady Pikovsky, “Hyperbolic chaos at blinking coupling of noisy oscillators”, Phys. Rev. E, 87:3 (2013), 032912  crossref  isi  scopus
    15. O. B. Isaeva, A. S. Kuznetsov, S. P. Kuznetsov, “Giperbolicheskii khaos pri parametricheskikh kolebaniyakh struny”, Nelineinaya dinam., 9:1 (2013), 3–10  mathnet
    16. O.B.. Isaeva, A.S.. Kuznetsov, S.P.. Kuznetsov, “Hyperbolic chaos of standing wave patterns generated parametrically by a modulated pump source”, Phys. Rev. E, 87:4 (2013), 040901(R)  crossref  adsnasa  isi  elib  scopus
    17. P.V. Kuptsov, “Violation of hyperbolicity via unstable dimension variability in a chain with local hyperbolic chaotic attractors”, J. Phys. A, 46:25 (2013), 254016  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    18. V. S. Anishchenko, S. V. Astakhov, “Poincaré recurrence theory and its applications to nonlinear physics”, Phys. Usp., 56:10 (2013), 955–972  mathnet  crossref  crossref  adsnasa  isi  elib
    19. Vyacheslav P. Kruglov, Sergey P. Kuznetsov, Arkady Pikovsky, “Attractor of SmaleWilliams Type in an Autonomous Distributed System”, Regul. Chaotic Dyn., 19:4 (2014), 483–494  mathnet  crossref  mathscinet  zmath
    20. A. V. Makarenko, “Analysis of the time structure of synchronization in multidimensional chaotic systems”, J. Exp. Theor. Phys, 120:5 (2015), 912  crossref  isi  elib  scopus
    21. Isaeva O.B., Kuznetsov S.P., Sataev I.R., Savin D.V., Seleznev E.P., “Hyperbolic Chaos and Other Phenomena of Complex Dynamics Depending on Parameters in a Nonautonomous System of Two Alternately Activated Oscillators”, Int. J. Bifurcation Chaos, 25:12 (2015), 1530033  crossref  mathscinet  zmath  isi  scopus
    22. Grishin S.V., Golova T.M., Morozova M.A., Romanenko D.V., Seleznev E.P., Sysoev I.V., Sharaevskii Yu.P., “Chaotic Parametric Soliton-Like Pulses in Ferromagnetic-Film Active Ring Resonators”, J. Exp. Theor. Phys., 121:4 (2015), 623–635  crossref  isi  elib  scopus
    23. 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  mathnet  crossref  mathscinet  adsnasa
    24. Sergeyev D., Barmina A., Zhanturina N., Shunkeyev K., “Dynamic Chaos in a Josephson Junction With An Anharmonic Current-Phase Relation”, 2015 International Siberian Conference on Control and Communications (Sibcon), IEEE, 2015  isi
    25. 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  mathnet  crossref  mathscinet
    26. Kuznetsov S.P., “From Geodesic Flow on a Surface of Negative Curvature to Electronic Generator of Robust Chaos”, Int. J. Bifurcation Chaos, 26:14 (2016), 1650232  crossref  mathscinet  zmath  isi  scopus
    27. Kuptsov P.V., Kuznetsov S.P., “Numerical test for hyperbolicity of chaotic dynamics in time-delay systems”, Phys. Rev. E, 94:1 (2016), 010201  crossref  isi  elib  scopus
    28. Makarenko A.V., “TQ-bifurcations in discrete dynamical systems: Analysis of qualitative rearrangements of the oscillation mode”, J. Exp. Theor. Phys., 123:4 (2016), 666–676  crossref  isi  elib  scopus
    29. Cherkasskii M.A., Nikitin A.A., Kalinikos B.A., “Theory of multinonlinear media and its application to the soliton processes in ferrite–ferroelectric structures”, J. Exp. Theor. Phys., 122:4 (2016), 727–733  crossref  isi  elib  scopus
    30. Makarenko A.V., “Analysis of phase synchronization of chaotic oscillations in terms of symbolic CTQ-analysis”, Tech. Phys., 61:2 (2016), 265–273  crossref  isi  elib  scopus
    31. Shuman V.N., “on Prognostication Ability of Active Geosystems: Metastability and Steady Transitions Instead of Attractors”, Geofiz. Zhurnal, 38:6 (2016), 3–24  crossref  isi
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    34. Kuptsov P.V., Kuznetsov S.P., “Numerical Test For Hyperbolicity in Chaotic Systems With Multiple Time Delays”, Commun. Nonlinear Sci. Numer. Simul., 56 (2018), 227–239  crossref  mathscinet  isi  scopus
    35. Xu M., Paul M.R., “Spatiotemporal Dynamics of the Covariant Lyapunov Vectors of Chaotic Convection”, Phys. Rev. E, 97:3 (2018), 032216  crossref  mathscinet  isi  scopus
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    37. Belik P., Dahl B., Dokken D., Potvin C.K., Scholz K., Shvartsman M., “Possible Implications of Self-Similarity For Tornadogenesis and Maintenance”, AIMS Math., 3:3 (2018), 365–390  crossref  isi
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