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ISSN 2542-1905 (Online)

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Kuznetsov S. P. Simple electronic chaos generators and their circuit simulation. Izvestiya VUZ. Applied Nonlinear Dynamics, 2018, vol. 26, iss. 3, pp. 35-61. DOI: 10.18500/0869-6632-2018-26-3-35-61

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Simple electronic chaos generators and their circuit simulation

Kuznetsov Sergey Petrovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences

Topic and aim. The aim of the work is to review circuits of chaos generators, those described in the literature and some original ones, in a unified style basing on circuit simulations with the NI Multisim package, which makes the comparison of the various devices apparent. Investigated models. A number of electronic chaos generators are considered including the Kolpitz oscillator, the Hartley oscillator, the RC chaos generator, variants of Chua circuit, the designs proposed by the Lithuanian group, Lorenz analog oscillator, generators of hyperbolic chaos with excitation transfer between alternately excited oscillators, as well as a ring generator with delayed feedback. Results. The circuit diagrams of chaos generators are presented, the principles of their operation are discussed, and circuit simulations are carried out using the NI Multisim package. For all considered systems the chaotic dynamics are illustrated consistently by waveforms of the signals, phase portraits of the attractors, spectra of the oscillations. Specially outlined are generators of robust chaos including the electronic analog of the Lorenz model and the circuits with Smale–Williams hyperbolic attractors, which seem preferable for possible applications due to low sensitivity of the chaos characteristics to parameter variations, manufacturing imperfections, interferences, etc. Discussion. The circuits collected in the paper correspond to low-frequency devices, but some of them may be useful in development of chaos generators also at high and ultrahigh frequencies. The material presented may be of interest for setting up laboratory and computer practical courses aimed at training specialists in the field of electronics and nonlinear dynamics, as well as for researchers interested in constructing chaos generators and their practical applications.

  1. Electronics of Backward-Wave Tubes / Eds Shevchik V.N., Trubetskov D.I. Saratov: Saratov University, 1975 (in Russian.)
  2. Bezruchko B.P., Kuznetsov S.P., Trubetskov D.I. Experimental observation of stochastic self-oscillations in the electron beam – backscattered electromagnetic wave dynamic system. JETP Lett, 1979, vol. 29, no. 3, pp. 162–165.
  3. Rabinovich M.I., Trubetskov D.I. Oscillations and waves: in linear and nonlinear systems. Springer Science & Business Media, 2012.
  4. Trubetskov D.I., Hramov A.E. Lectures on Microwave Electronics for Physicists. Moscow: Fizmatlit, 2003. (In Russian.)
  5. Dmitriev A.S., Efremova E.V., Maksimov N.A., Panas A.I. Generation of chaos. Moscow, Technosfera, 2012. 432 p. (In Russian.)
  6. Dmitriev A.S., Panas A.I. Dynamic Chaos: New Information Carriers for Communications Systems. Moscow: Fizmatlit, 2002. (In Russian.)
  7. Myasin Е.А. Investigations of the hf noise generation in IRE of Academy of Sciences of USSR at 1962–1967 years – The beginning of the new science direction. Izvestiya VUZ, Applied Nonlinear Dynamics, 2014, vol. 22, no. 1, pp. 104–122. (In Russian.)
  8. Lukin K.A. Noise radar technology //Telecommunications and Radio Engineering. 2001. Vol. 55, № 12. Pp. 8–16.
  9. Stojanovski T., Kocarev L. Chaos-based random number generators – Part I: Analysis //IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications. 2001. Vol. 48, № 3. Pp. 281–288.
  10. Stojanovski T., Pihl J., Kocarev L. Chaos-based random number generators – Part II: Practical realization //IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications. 2001. Vol. 48, № 3. Pp. 382–385.
  11. Baptista M. Cryptography with chaos //Physics Letters A. 1998. Vol. 240, № 1–2. Pp. 50–54.
  12. Herniter M.E. Schematic Capture with Multisim. Prentice Hall, 2004.
  13. Zeraoulia E., Sprott J.C. Robust Chaos and its Applications. Singapore: World Scientific, 2012.
  14. Kuznetsov S.P. Dynamical chaos and uniformly hyperbolic attractors: from mathematics to physics. Physics-Uspekhi, 2011, vol. 54, no. 2, pp. 119–144.
  15. Gonchenko A.S., Gonchenko S.V., Kazakov A.O., Кozlov A.D. Mathematical theory of dynamical chaos and its applications: Review. Part 1. Pseudohyperbolic attractors. Izvestiya VUZ, Applied Nonlinear Dynamics, 2017, vol. 25, no. 2, pp. 4–36. (In Russian.)
  16. Lorenz E.N. Deterministic nonperiodic flow //Journal of the Atmospheric Sciences. 1963. Vol. 20, № 2. Pp. 130–141.
  17. Sparrow C. The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors. Springer Science & Business Media, 2012.
  18. Kuznetsov S.P. Dynamical chaos. M., Fizmatlit, 2001. (In Russian.) 
  19. Shilnikov L. Mathematical problems of nonlinear dynamics: A tutorial //International Journal of Bifurcation and Chaos. 1997. Vol. 7, № 9. Pp. 1953–2001.
  20. Kennedy M.P. Chaos in the Colpitts oscillator //IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications. 1994. Vol. 41, № 11. Pp. 771–774.  
  21. Peter K. Chaos in Hartley’s oscillator //International Journal of Bifurcation and Chaos. 2002. Vol. 12, № 10. Pp. 2229–2232.
  22. Keuninckx L., Van der Sande G., Danckaert J. Simple two-transistor single-supply resistor–capacitor chaotic oscillator //IEEE Transactions on Circuits and Systems II: Express Briefs. 2015. Vol. 62, № 9. Pp. 891–895.
  23. Chua L.O., Wu C.W., Huang A., Zhong G.Q. A universal circuit for studying and generating chaos. I. Routes to chaos //IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications. 1993. Vol. 40, № 10. Pp. 732–744.
  24. Kennedy M.P. Robust op amp realization of Chua’s circuit // Frequenz. Journal of RF-Engineering and Telecommunications. 1992. Vol. 46, № 3–4. Pp. 66–80.
  25. Morgul O. Wien bridge based RC chaos generator // Electronics Letters. 1995. Vol. 31, № 24. Pp. 2058–2059.
  26. Tamasevicius A., Mykolaitis G., Pyragas V., Pyragas K. A simple chaotic oscillator for educational purposes // European Journal of Physics. 2004. Vol. 26, № 1. Pp. 61–63.
  27. Namajunas A., Tamasevicius A. Simple RC chaotic oscillator //Electronics Letters. 1996. Vol. 32, № 11. Pp. 945–946.
  28. Tamasevicius A., Bumeliene S., Kirvaitis R., Mykolaitis G., Tamaseviciute E. Autonomous Duffing–Holmes type chaotic oscillator // Elektronika ir Elektrotechnika. 2009. Vol. 93, № 5. Pp. 43–46.
  29. Oraevskii A.N. Masers, lasers, and strange attractors. Quantum electronics, 1981, vol. 11, no. 1, pp. 71–78.
  30. Kolar M., Gumbs G. Theory for the experimental observation of chaos in a rotating waterwheel // Physical Review A. 1992. Vol. 45, № 2. Pp. 626–637.
  31. Glukhovskii A.B. Nonlinear systems that are superpositions of gyrostats. Soviet Physics Doklady, 1982, vol. 27, pp. 823–827.
  32. Doroshin A. V. Modeling of chaotic motion of gyrostats in resistant environment on the base of dynamical systems with strange attractors // Communications in Nonlinear Science and Numerical Simulation. 2011. Vol. 16, № 8. Pp. 3188–3202.
  33. Kuznetsov S.P. Lorenz type attractor in electronic parametric generator and its transformation outside the accurate parametric resonance. Izvestiya VUZ, Applied Nonlinear Dynamics, 2016, vol. 24, no. 3, pp. 68–87.
  34. Horowitz P. Build a Lorenz attractor:
  35. Kuznetsov S.P. Example of a physical system with a hyperbolic attractor of the Smale–Williams type // Physical Review Letters. 2005. Vol. 95, № 14. 144101.
  36. Kuznetsov S.P. Electronic circuits manifesting hyperbolic chaos and simulation of their dynamics using software package MULTISIM. Izvestiya VUZ, Applied Nonlinear Dynamics, 2011, vol. 19, no. 5, pp. 98–115.
  37. Kuznetsov S.P., Seleznev E.P. A strange attractor of the Smale–Williams type in the chaotic dynamics of a physical system. JETP, 2006, vol. 102, no. 2, pp. 355–364.
  38. 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 // International Journal of Bifurcation and Chaos. 2015. Vol. 25, № 12. Pp. 1530033.
  39. Kuznetsov A.P., Kuznetsov S.P., Pikovski A.S., Turukina L.V. Chaotic dynamics in the systems of coupling nonautonomous oscillators with resonance and nonresonance communicator of the signal. Izvestiya VUZ, Applied Nonlinear Dynamics, 2007,vol. 15, no. 6, pp. 75–85. (In Russian.)
  40. Kruglov V.P., Doroshenko V.M., Kuznetsov S.P. Hyperbolic chaos in the coupled Bonhoeffer – van der Pol oscillators operating with excitation of relaxation selfoscillations. Nonlinear waves – 2018. XVIII Scientific School. February 26 – March 4, 2018. Abstracts of reports of young scientists. Nizhny Novgorod: IAP RAS, 2018. P. 87–89.
  41. Kuznetsov S. P. Complex dynamics of oscillators with delayed feedback. Radiophysics and Quantum Electronics, 1982, vol. 25, no. 12, pp. 996–1009.
  42. Vallee R., Delisle C., Chrostowski J. Noise versus chaos in acousto-optic bistability. Physical Review A. 1984. Vol. 30, № 1. Pp. 336–342.
  43. Mackey M.C., Glass L. Oscillation and chaos in physiological control systems // Science, 1977. Vol. 97, №4 300. Pp. 287–289.
  44. Hu H.Y. and Wang Z.H. Dynamics of Controlled Mechanical Systems with Delayed Feedback. Springerб 2002.
  45. Chiasson J.N. and Loiseaum J.J. (eds.): Applications of Time Delay Systems. Springer, 2007.
  46. Kislov V.Ya., Zalogin N.N., Myasin E.A. Study of stochastic self-oscillating processes in self-excited oscillators with delay. Radiotekhnika i elektronika, 1979, vol. 24, no. 6, pp. 1118–1130. (In Russian.)
  47. Ikeda K., Daido H., Akimoto O. Optical turbulence: chaotic behavior of transmitted light from a ring cavity //Physical Review Letters. 1980. Vol. 45, № 9. Pp. 709–712.
  48. Farmer J.D. Chaotic attractors of an infinite-dimensional dynamical system // Physica D: Nonlinear Phenomena. 1982. Vol. 4, № 3. Pp. 366–393.
  49. Chrostowski J., Vallee R., Delisle C. Self-pulsing and chaos in acoustooptic bistability // Canadian Journal of Physics. 1983. Vol. 61, № 8. Pp. 1143–1148.
  50. Chevalier T., Freund A., Ross J. The effects of a nonlinear delayed feedback on a chemical reaction // The Journal of Chemical Physics. 1991. Vol. 95, № 1. С. 308–316.  
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