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UFN, 2005, Volume 175, Number 2, Pages 163–179 (Mi ufn147)  

This article is cited in 28 scientific papers (total in 28 papers)

REVIEWS OF TOPICAL PROBLEMS

Statistical properties of dynamical chaos

V. S. Anishchenko, T. E. Vadivasova, G. A. Okrokvertskhov, G. I. Strelkova

Department of Physics, Institute of Nonlinear Dynamics, Saratov State University

Abstract: Statistical descriptions of dynamical chaos and investigations of noise effects on chaotic oscillation regimes are reviewed. Nearly hyperbolic and nonhyperbolic chaotic attractors are studied. An illustration of the technique of diagnosing the attractor type in numerical simulations is given. Regularities in relaxation to the invariant probability distribution are analyzed for various types of attractors. Spectral-correlative properties of chaotic oscillations are investigated. Decay laws for the autocorrelation functions and the shapes of the power spectra are found, along with their relationship to the Lyapunov exponents, diffusion of the instantaneous phase, and the intensity of external noise. The mechanism of the onset of chaos and its relationship to the characteristics of the spiral attractors are demonstrated for inhomogeneous media that can be modeled by the Ginzburg – Landau equation. Numerical data are compared with experimental results.

DOI: https://doi.org/10.3367/UFNr.0175.200502c.0163

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English version:
Physics–Uspekhi, 2005, 48:2, 151–166

Bibliographic databases:

PACS: 02.50.-r, 05.45.-a, 47.52.+j
Received: September 17, 2004

Citation: V. S. Anishchenko, T. E. Vadivasova, G. A. Okrokvertskhov, G. I. Strelkova, “Statistical properties of dynamical chaos”, UFN, 175:2 (2005), 163–179; Phys. Usp., 48:2 (2005), 151–166

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. V. S. Anishchenko, M. V. Anufrieva, T. E. Vadivasova, “Stochastic resonance in a bistable system driven by a chaotic signal”, Tech Phys Lett, 32:10 (2006), 873  crossref  adsnasa  isi  elib  scopus
    2. A. A. Koronovskii, M. K. Kurovskaya, O. I. Moskalenko, A. E. Hramov, “Two scenarios of breaking chaotic phase synchronization”, Tech Phys, 52:1 (2007), 19  crossref  mathscinet  isi  elib  scopus
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    4. Alexey Yu. Jalnine, Sergey P. Kuznetsov, “Effect of noise in a nonautonomous system of alternately excited oscillators with a hyperbolic strange attractor”, Phys Rev E, 77:3 (2008), 036220  crossref  mathscinet  isi  scopus
    5. Anishchenko, VS, “RELATIVE KOLMOGOROV ENTROPY OF A CHAOTIC SYSTEM IN THE PRESENCE OF NOISE”, International Journal of Bifurcation and Chaos, 18:9 (2008), 2851  crossref  mathscinet  adsnasa  isi  scopus
    6. Pavlov, AN, “Analysis of correlation properties of random processes using short signals”, Technical Physics Letters, 34:4 (2008), 306  crossref  mathscinet  adsnasa  isi  elib  scopus
    7. Dmitry A. Smirnov, Boris P. Bezruchko, “Detection of couplings in ensembles of stochastic oscillators”, Phys Rev E, 79:4 (2009), 046204  crossref  mathscinet  isi  scopus
    8. S. D. Glyzin, “Raznostnye approksimatsii uravneniya «reaktsiya–diffuziya» na otrezke”, Model. i analiz inform. sistem, 16:3 (2009), 96–115  mathnet
    9. Kontorovich V., Lovtchikova Z., “Non Linear Filtering Algorithms For Chaotic Signals: A Comparative Study”, Second International Workshop on Nonlinear Dynamics and Synchronization, Smart System Technologies, 4, 2009, 221–227  isi
    10. C. Komalapriya, M. C. Romano, M. Thiel, N. Marwan, J. Kurths, I. Z. Kiss, J. L. Hudson, “An automated algorithm for the generation of dynamically reconstructed trajectories”, Chaos, 20:1 (2010), 013107  crossref  adsnasa  isi  scopus
    11. Kontorovich V., Lovtchikova Z., Ramos-Alarcon F., “Correlation Properties of Chaos: Cumulant Approach”, Mathematical & Computational Applications, 15:5 (2010), 946–952  crossref  mathscinet  isi
    12. V. S. Malyaev, T. E. Vadivasova, “Otsenka parametrov zashumlennykh dinamicheskikh sistem”, Nelineinaya dinam., 6:2 (2010), 267–276  mathnet
    13. A. Yu. Loskutov, “Fascination of chaos”, Phys. Usp., 53:12 (2010), 1257–1280  mathnet  crossref  crossref  isi  elib
    14. Piterbarg L.I., “Parameter estimation from small biased samples: Fuzzy sets vs statistics”, Fuzzy Sets and Systems, 170:1 (2011), 1–21  crossref  mathscinet  zmath  isi  scopus
    15. Anderson R., Milutinovic D., “A Stochastic Approach to Dubins Feedback Control for Target Tracking”, IEEE/RSJ International Conference on Intelligent Robots and Systems, IEEE International Conference on Intelligent Robots and Systems, 2011, 3917–3922  crossref  isi
    16. 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
    17. K. A. Rybkin, “Flikker-shum pri svobodnom padenii tsilindrov v vozdukhe”, Nelineinaya dinam., 8:3 (2012), 629–639  mathnet
    18. Kontorovich V., Lovtchikova Z., “Multi-Moment Statistical Characterization and Nonlinear Filtering of Chaos”, Compel-Int. J. Comp. Math. Electr. Electron. Eng., 32:3 (2013), 885–900  crossref  mathscinet  zmath  isi  scopus
    19. D.A.. Smirnov, E.V.. Sidak, B.P.. Bezruchko, “Detection of coupling between oscillators with analytic tests for significance”, Eur. Phys. J. Spec. Top, 222:10 (2013), 2441  crossref  isi  scopus
    20. Ya. I. Boev, N. I. Biryukova, V. S. Anischenko, “Statistika vremen vozvrata Puankare v neavtonomnom odnomernom khaoticheskom otobrazhenii”, Nelineinaya dinam., 10:1 (2014), 3–16  mathnet
    21. Yaroslav Boev, Nadezhda Semenova, Galina Strelkova, Vadim Anishchenko, “Poincaré Recurrences in a Nonautonomous Chaotic Map”, Int. J. Bifurcation Chaos, 24:08 (2014), 1440016  crossref  mathscinet  zmath  isi  scopus
    22. Jaykov Foukzon, “New Scenario for Transition to Slow 3-D Turbulence”, JAMP, 03:03 (2015), 371  crossref
    23. 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
    24. Sidak E.V., Smirnov D.A., Osipov G.V., Bezruchko B.P., “Influence of nonlinear amplitude dynamics on estimated delay time of coupling between self-oscillatory systems”, Tech. Phys. Lett., 42:3 (2016), 287–290  crossref  isi  elib  scopus
    25. Semenov A., “the Additive White Gaussian Noise Impact on the Deterministic Chaos Oscillator Based on a Field-Effect Transistor Structure With Negative Resistance”, 2016 International Conference Radio Electronics & Info Communications (Ukrmico), IEEE, 2016  isi
    26. Sidak E.V., Smirnov D.A., Bezruchko B.P., “Estimation of the time delay of coupling between oscillators from time realizations of oscillation phases for different properties of phase dynamics”, J. Commun. Technol. Electron., 62:3 (2017), 241–250  crossref  isi  scopus
    27. A. V. Makarenko, “Metrization of the T-alphabet: measuring the distance between multidimensional real discrete sequences”, Autom. Remote Control, 80:1 (2019), 138–149  mathnet  crossref
    28. Glyzin S.D., “Difference Approximations of a Reaction-Diffusion Equation on Segments”, Autom. Control Comp. Sci., 52:7 (2018), 762–776  crossref  isi  scopus
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