ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

For citation:

Kuznetsov S. P., Ponomarenko V. I., Seleznev E. P. Autonomous system generating hyperbolic chaos: circuit simulation and experiment. Izvestiya VUZ. Applied Nonlinear Dynamics, 2013, vol. 21, iss. 5, pp. 17-30. DOI: 10.18500/0869-6632-2013-21-5-17-30

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text PDF(Ru):
PDF icon 2013no5p017.pdf
(downloads: 145)
Language: 
Russian
Heading: 
Applied Problems of Nonlinear Oscillation and Wave Theory
Article type: 
Article
UDC: 
517.9:537.86:621.373
DOI: 
10.18500/0869-6632-2013-21-5-17-30

Autonomous system generating hyperbolic chaos: circuit simulation and experiment

Autors: 
Kuznetsov Sergey Petrovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Ponomarenko Vladimir Ivanovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Seleznev Evgeny Petrovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Abstract: 

We consider an electronic device, which represents an autonomous dynamical system with hyperbolic attractor of the Smale–Williams type in the Poincare map. Simulation of chaotic dynamics in the software environment Multisim has been undertaken. The generator of hyperbolic chaos is implemented as a laboratory model; its experimental investigation is carried out, and good compliance with the observed dynamics in the numerical and circuit simulation has been demonstrated. A distinctive feature of the system is attractiveness for potential applications due to its robustness, or structural stability, i.e. insensitivity of the generated chaos characteristics in respect to noises, technical fluctuations, fabrication imperfections etc.

Key words: 
chaos
Attractor
Smale–Williams solenoid
autonomous system
experiment
operational amplifier
Multisim
Reference: 
  1. Smale S. Differentiable dynamical systems. Uspekhi Mat. Nauk. 1970;25(1):113–185.
  2. Anosov DV. et al. Dynamical systems with hyperbolic behavior. Dynamical systems – 9, Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Fund. Napr. 66. Moscow: VINITI; 1991. 5–242 p.
  3. Sinai YG. Stochasticity of dynamic systems. Nonlinear waves. Ed. Gapanov-Grehov AV. Moscow: Nauka; 1979. 361 p. (In Russian).
  4. Shilnikov L. Mathematical problems of nonlinear dynamics: A tutorial. Int. J. of Bifurcation and Chaos. 1997;7(9):1353–2001.
  5. Kuznetsov SP. Example of a physical system with a hyperbolic attractor of the Smale–Williams type. Phys. Rev. Lett. 2005;95(14):144101. DOI: 10.1103/PhysRevLett.95.144101.
  6. Kuznetsov SP, Pikovsky A. Autonomous coupled oscillators with hyperbolic strange attractors. Physica D. 2007;232(2):87–102. DOI: 10.1016/J.PHYSD.2007.05.008.
  7. Kuznetsov SP. Dynamical chaos and uniformly hyperbolic attractors: from mathematics to physics. Phys. Usp. 2011;54(2):119–144. DOI: 10.3367/UFNr.0181.201102a.0121.
  8. Kuznetsov SP. Hyperbolic Chaos: A Physicist’s View. Beijing: Higher Education Press, Berlin, Heidelberg: Springer–Verlag; 2012. 336 p.
  9. Dmitriev AS, Panas AI. Dynamic chaos: New media for communication systems. Moscow: Fizmatlit. 2002. 252 p. (In Russian).
  10. Koronovskii AA, Moskalenko OI, Hramov AE. On the use of chaotic synchronization for secure communication. Phys. Usp. 2009;52(12):1213–1238. DOI: 10.3367/UFNr.0179.200912c.1281.
  11. Lukin KA. Noise radar technology. Telecommunications and Radio-Engineering. 2001;16(12):8–16.
  12. Baptista MS. Cryptography with chaos. Physics Letters A. 1998;240(1-2):50–54. DOI:10.1016/S0375-9601(98)00086-3.
  13. Ptitsin NV. Application of the theory of deterministic chaos in cryptography. Moscow: BMSTU Press; 2002. 81 p. (In Russian).
  14. Stojanovski T, Kocarev L. Chaos-based random number generators: Part I and II. IEEE Trans. Circuits and Systems. 2001;48(3):281–288. DOI:10.1109/81.915385.
  15. Andronov AA, Witt AA, Haikin SE. Oscillation theory. Moscow: Nauka; 1959. 915 p. (In Russian).
  16. Rabinovich MI, Trubetskov DI. Introduction to the theory of oscillations and waves. Moscow: Nauka; 1984. 432 p. (In Russian).
  17. Elhadj Z, Sprott JC. Robust Chaos and Its Applications. Singapore: World Scientic; 2011. 472 p.
  18. Kuznetsov SP, Seleznev EP. A strange attractor of the Smale-Williams type in the chaotic dynamics of a physical system. Journal of Experimental and Theoretical Physics. 2006;102(2):355–364. DOI: 10.1134/S1063776106020166.
  19. Kuznetsov SP, Ponomarenko VI. Realization of a strange attractor of the Smale-Williams type in a radiotechnical delay-feedback oscillator. Technical Physics Letters. 2008;34(9):771–773. DOI: 10.1134/S1063785008090162.
  20. Baranov SV, Kuznetsov SP, Ponomarenko VI. Chaos in the phase dynamics of q-switched van der Pol oscillator with additional delayed feedback loop. Izvestiya VUZ. Applied Nonlinear Dynamics. 2010;18(1):12–23. DOI: 10.18500/0869-6632-2010-18-1-12-23.
  21. Kuznetsov SP. Plykin type attractor in electronic device simulated in MULTISIM. CHAOS. 2011;21(4):043105. DOI:10.1063/1.3646903.
  22. Kuznetsov SP. Electronic circuits manifesting hyperbolic chaos and simulation of their dynamics using software package Multisim. Izvestiya VUZ. Applied Nonlinear Dynamics. 2011;19(5):98–115. DOI: 10.18500/0869-6632-2011-19-5-98-115.
  23. Kuznetsov SP. Dynamic chaos. Moscow: Физматлит; 2001. 296 p. (In Russian).
  24. Horowitz P, Hill W. The art of electronics. Moscow: Mir; 1986. Vol. 1. 510 p., Vol. 2. 592 p.
  25. Rempen IS, Egorov EN, Savin AN, Ponomarenko VI. Operational amplifiers. Teaching and methodological manual. Saratov: Kolledg; 2004. Vol. 1. 19 p., Vol. II. 16 p. (In Russian).
  26. Makarenko VV. Simulation of electronic devices using the NI MULTISIM program. Electronic modeling Electronic components and systems. Kiev: VD MAIS; 2008. № 1, 50–56; № 2, 51–57; № 3, 44–51; № 4, 44–51, № 6, 46–53; № 7, 54–59; № 8, 46–56; № 9, 65–69; № 12, 47–52.
  27. Varzarev YN, Ivantsov VV, Spyridonov BG. Modeling of electronic diagrams in Multisim system. Taganrog: SfedU Press; 2008. 81 p. (In Russian).
Received: 
27.02.2013
Accepted: 
27.02.2013
Published: 
31.12.2013
Short text (in English):
PDF icon a2013no5p017.pdf
(downloads: 105)
  • 1885 reads