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

For citation:

Kuznetsov S. P., Sokha Y. I. Hyperchaos in model nonautonomous system with a cascade excitation transmission through the spectrum. Izvestiya VUZ. Applied Nonlinear Dynamics, 2010, vol. 18, iss. 3, pp. 24-32. DOI: 10.18500/0869-6632-2010-18-3-24-32

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):
(downloads: 114)
Article type: 

Hyperchaos in model nonautonomous system with a cascade excitation transmission through the spectrum

Kuznetsov Sergey Petrovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Sokha Yury Ivanovich, Saratov State University

One of the key turbulence theory idea is a cascade energy transmission through the spectrum from large to small scales. It appears that this idea could be used for complex dynamics realization in a different-nature systems even when equations are knowingly differ from hydrodynamical. The system of four van der Pol oscillators is considered in this paper. Chaos generation is realized by cascade excitation transmission from one oscillator to another with frequency doubling. Due to slow forced modulation of the parameters responsible for the self-excitation two pair of oscillators become active turn by turn. In the beginning of each new active stage the excitation of oscillators from second to fourth are stimulated by oscillators with the half frequencies through quadratic nonlinear element. Excitation from the last oscillator to the first one is transmitted by the signal accepted via quadratic nonlinearity in the presence of auxiliary harmonic signal. In accordance with the results of numeric investigation the two positive Lyapunov exponents hyperchaos mode takes place.

  1. Landau LD, Lifshitz EM. Hydrodynamics. Moscow: Nauka; 1986. 736 p. (in Russian).
  2. Monin AS, Yaglom AM. Statistical hydromechanics. Part 1. Moscow: Nauka; 1965. 640 p. Part 2. Moscow: Nauka; 1967. 720 p. (in Russian).
  3. 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.
  4. 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.
  5. Kuznetsov AP, Sataev IR. Verification of hyperbolicity conditions for a chaotic attractor in a system of coupled nonautonomous van der Pol oscillators. Izvestiya VUZ. Applied Nonlinear Dynamics. 2006;14(5):3–29 (in Russian). DOI: 10.18500/0869-6632-2006-14-5-3-29.
  6. Isaeva OB, Jalnine AY, Kuznetsov SP. Arnold’s cat map dynamics in a system of coupled nonautonomous van der Pol oscillators. Phys. Rev. E. 2006;74:046207. DOI: 10.1103/PhysRevE.74.046207.
  7. Kuznetsov SP, Pikovsky A. Autonomous coupled oscillators with hyperbolic strange attractors. Physica D. 2007;232:87–102. DOI: 10.1016/j.physd.2007.05.008.
  8. Kuznetsov CP. On the implementation of some classical models and phenomena of nonlinear dynamics based on coupled non-autonomous oscillators In: Nonlinear Waves-2006. Ed. Gaponov-Grekhov AV, Nekorkin VI. N. Novgorod: Institute of Applied Physics RAS. 2007:68–84 (in Russian).
  9. Kuznetsov AP, Kuznetsov SP, Pikovsky AS, Turukina LV. Chaotic dynamics in the systems of coupling nonautonomous oscillators with resonance and nonresonance communicator of the signal. Izvestiya VUZ. Applied Nonlinear Dynamics. 2007;15(6):75–85 (in Russian). DOI: 10.18500/0869-6632-2007-15-6-75-85.
  10. Jalnine AYu, Kuznetsov SP. On the realization of the Hunt-Ott strange nonchaotic attractor in a physical system. Technical Physics. 2007;52(4):401–408. DOI: 10.1134/S1063784207040020.
  11. Isaeva OB, Kuznetsov SP, Osbaldestin A. Complex analytic dynamics phenomena in a system of coupled nonautonomous oscillators with alternative excitation. Technical Physics Letters. 2007;33(9):748–751. DOI: 10.1134/S1063785007090106.
  12. Rossler OE. An equation for hyperchaos. Phys. Lett. A. 1979;71(2–3):155–157. DOI: 10.1016/0375-9601(79)90150-6.
  13. Benettin G, Galgani L, Giorgilli A, Strelcyn JM. Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems: A method for computing all of them. Part I: Theory. Meccanica. 1980;15:9–20. Part II: Numerical application. Meccanica. 1980;15:21–30. DOI: 10.1007/BF02128237.
  14. Kaplan JL, Yorke JA. Lecture Notes in Mathematics. Berlin: Springer-Verlag; 1979;730. 204 p.
  15. Sveshnikov AA. Applied Methods of the Theory of Random Functions. Moscow: Nauka, Fizmatlit; 1968. 464 p. (in Russian).
Short text (in English):
(downloads: 96)