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

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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, iss. 3, pp. 68-87. DOI: 10.18500/0869-6632-2016-24-3-68-87

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Language: 
Russian
Heading: 
Methodical Papers on Nonlinear Dynamics
Article type: 
Article
UDC: 
621.373.7
DOI: 
10.18500/0869-6632-2016-24-3-68-87

Lorenz type attractor in electronic parametric generator and its transformation outside the accurate parametric resonance

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

The paper deals with a parametric oscillator composed of three LC-circuits and a quadratic nonlinear reactive element built on the basis of an operational amplifier and an analog multiplier; the equations for amplitudes of the interacting modes are derived. Motivation is a desire to implement the mechanism of parametric interaction of oscillatory modes giving rise to emergence of a strange attractor of Lorenz type without distortions introduced by nonlinearities of order three and higher. The study is based on a combination of circuit simulation using Multisim software and numerical integration of the dynamic equations of the system both in its original form and in the form of reduced equations for the slowly varying complex amplitudes. The proposed scheme for the first time allows demonstrating the decay mechanism of chaos generation described earlier by Pikovsky, Rabinovich and Trahtengerts in concern to the waves in magnetized plasma, in an electronic device in purified form. In addition to observation of the Lorenz-type attractor and characteristic features of the respective dynamics by means of the circuit simulation and on the basis of numerical integration of equations in the case of precise parametric resonance conditions, a study of transformation of the attractors is carried out with detuning frequencies, and the corresponding chart of dynamical regimes on the parameter plane is presented. It is shown that instead of the quasi-hyperbolic Lorenz-type attractor, with frequencies deviating from the exact parametric resonance, distinct types of attractors arise, although similar in shape to the original one, but lacking robustness: under variations of the parameters chaos may disappear with emergence of regular oscillatory regimes. 

Key words: 
Parametric oscillator
the Lorenz attractor
analog simulation
Reference: 
  1. Pikovski A.S., Rabinovich M.I., Trakhtengerts V.Y. Appearance of chaos at decay saturation of parametric instability // Sov. Phys. JETP. 1978. Vol. 47. P. 715–719.
  2. Louisell W.H. Coupled Mode and Parametric Electronics. Wiley: New York, 1960. 268 p.
  3. Akhmanov S.A., Khokhlov R.V. Parametric amplifiers and generators of light // Physics-Uspekhi. 1966. Vol. 9, No 2. P. 210–222.
  4. Ostrovskii L.A., Papilova I.A., Sutin A.M. Parametric ultrasound generator // Soviet Journal of Experimental and Theoretical Physics Letters. 1972. Vol. 15. P. 322–323 (in Russian).
  5. Akulenko L.D. Parametric control of the oscillations and rotations of a physical pendulum (swing) // Prikladnaya Matematika i Mekhanika. 1993. Vol. 57, No 2. P. 82–91 (in Russian).
  6. Lorenz E.N. Deterministic nonperiodic flow // Journal of the Atmospheric Sciences. 1963. Vol. 20, No 2. P. 130–141.
  7. Sparrow C. The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors. NY, Heidelberg, Berlin: Springer-Verlag, 1982. 270 p.
  8. Kuznetsov S.P. Dynamic Chaos. 2nd ed. Fizmatlit: Moscow, 2006. 356 p. (in Russian).
  9. Anishchenko V.S. Attractors of Dynamical Systems // Izvestiya VUZ. Applied Nonli-near Dynamics. 1997. Vol. 5, No 1. P. 109–127 (in Russian).
  10. Bonatti C., Diaz L.J., Viana M. Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probobalistic Perspective. Encyclopedia of Mathematical Sciences. Vol. 102. Springer: Berlin, Heidelberg, New-York, 2005. 384 p.
  11. Banerjee S., Yorke J.A., Grebogi C. Robust chaos // Phys. Rev. Lett. 1998. Vol. 80. P. 3049–3052.
  12. Elhadj Z. and Sprott J.C. Robust Chaos and its Applications. Singapore: World Scientific, 2011. 454 p.
  13. Oraevskii A.N. Masers, lasers, and strange attractors // Quantum Electronics. 1981. Vol. 11, No 1. P. 71–78.
  14. Oraevsky A.N. Dynamics of single-mode lasers and dynamical chaos // Izvestiya VUZ. Applied Nonlinear Dynamics. 1996. Vol. 4, No 1. P. 3–13 (in Russian).
  15. Haken H. Analogy between higher instabilities in fluids and lasers // Physics Letters A. 1975. Vol. 53, No 1. P. 77–78.
  16. Kolar M., Gumbs G. Theory for the experimental observation of chaos in a rotating waterwheel // Physical Review A. 1992. Vol. 45, No 2. P. 626–637.
  17. Glukhovskii A.B. Nonlinear systems that are superpositions of gyrostats // Soviet Physics Doklady. 1982. Vol. 27. P. 823.
  18. 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, No 8. P. 3188–3202.
  19. Chen H.K., Lee C.I. Anti-control of chaos in rigid body motion //Chaos, Solitons &Fractals. 2004. Vol. 21, No 4. P. 957–965.
  20. Poland D. Cooperative catalysis and chemical chaos: a chemical model for the Lorenz equations // Physica D: Nonlinear Phenomena. 1993. Vol. 65, No 1. P. 86–99.
  21. Cuomo K.M., Oppenheim A.V. Circuit implementation of synchronized chaos with applications to communications // Phys. Rev. Lett. 1993. Vol. 71, No 1. P. 65–68.
  22. Peters F., Lobry L., Lemaire E. Experimental observation of Lorenz chaos in the Quincke rotor dynamics // Chaos: An Interdisciplinary Journal of Nonlinear Science. 2005. Vol. 15, No 1. P. 013102.
  23. Rucklidge A.M. Chaos in magnetoconvection // Nonlinearity. 1994. Vol. 7, No 6. P. 1565–1591.
  24. Hemail N. Strange attractors in brushless DC motor //IEEE Transactions on Circuits and System-I: Fundamental Theory and Application. 1994. Vol. 41, No 1. P. 40–45.
  25. Gibbon J.D., McGuinness M.J. The real and complex Lorenz equations in rotating fluids and lasers // Physica D: Nonlinear Phenomena. 1982. Vol. 5, No 1. P. 108–122.
  26. Fowler A.C., Gibbon J.D., McGuinness M.J. The complex Lorenz equations // Physica D: Nonlinear Phenomena. 1982. Vol. 4, No 2. P. 139–163.
  27. Mahmoud G.M., Ahmed M.E., Mahmoud E.E. Analysis of hyperchaotic complex Lorenz systems // International Journal of Modern Physics C. 2008. Vol. 19, No 10. P. 1477–1494.
  28. Wang P.K.C., Masui K. Intermittent phase unlocking in a resonant three-wave interaction with parametric excitation // Physics Letters A. 1981. Vol. 81, No 2. P. 97–101.
  29. Letellier C., Aguirre L.A., Maquet J., Lefebvre B. Analogy between a 10D model for nonlinear wave–wave interaction in a plasma and the 3D Lorenz dynamics // Physica D: Nonlinear Phenomena. 2003. Vol. 179, No 1. P. 33–52.
  30. Llibre J., Messias M., da Silva P.R. On the global dynamics of the Rabinovich system // Journal of Physics A: Mathematical and Theoretical. 2008. Vol. 41, No 27. P. 275210.
  31. Kuznetsov S.P. Parametric chaos generator operating on a varactor diode with the instability limitation decay mechanism // Technical Physics. 2016. Vol. 61, No 3. P. 436–445.
  32. Liu Y., Yang Q., Pang G. A hyperchaotic system from the Rabinovich system // Journal of Computational and Applied Mathematics. 2010. Vol. 234, No 1. P. 101–113.
  33. Kuznetsov S.P. Dynamical chaos and uniformly hyperbolic attractors: from mathematics to physics // Physics-Uspekhi. 2011. Vol. 54, No 2. P. 119–144.
  34. Tucker W. A rigorous ODE solver and Smale’s 14th problem // Comp. Math. 2002. Vol. 2. P. 53–117.
  35. Benettin G., Galgani L., Giorgilli A., Strelcyn J.-M. Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems: A method for computing all of them // Meccanica. 1980. Vol. 15. P. 9–20.
  36. Henon M.  A two-dimensional mapping with a strange attractor // Commun. Math. Phys. 1976. Vol. 50. P. 69–77.
  37. Rossler O.E. Continuous chaos: four prototype equations // Ann. New York Academy of Sciences. 1979. Vol. 316. P. 376–392.
  38. Afraimovich V.S. Strange attractors and quasiattractors // Nonlinear and turbulent processes in physics. 1984. Vol. 1. P. 1133–1138.
  39. Shilnikov L.P. Bifurcations and strange attractors // Vestnik Nizhegorodskogo Universiteta. 2011, No 4(2). P. 364–366.
  40. Kozlov V.V. On the problem of fall of a rigid body in a resisting medium // Moscow University Mechanics Bulletin. 1990. Vol. 45, No 1. P. 30–35.
  41. Kuznetsov S.P. Plate falling in a fluid: Regular and chaotic dynamics of finitedimensional models // Regular and Chaotic Dynamics. 2015. Vol. 20, No 3. P. 345–382.
Received: 
30.06.2016
Accepted: 
30.06.2016
Published: 
30.06.2016
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