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

For citation:

Kuznetsov A. P., Popova E. S., Seleznev E. P., Stankevich N. V. Doubling and destruction of the tri-frequencies torus in the nonlinear oscillator under quasi-periodic exitation: experiment. Izvestiya VUZ. Applied Nonlinear Dynamics, 2013, vol. 21, iss. 5, pp. 31-39. DOI: 10.18500/0869-6632-2013-21-5-31-39

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 2013no5p031.pdf
(downloads: 132)
Language: 
Russian
Heading: 
Applied Problems of Nonlinear Oscillation and Wave Theory
Article type: 
Article
UDC: 
517.9
DOI: 
10.18500/0869-6632-2013-21-5-31-39

Doubling and destruction of the tri-frequencies torus in the nonlinear oscillator under quasi-periodic exitation: experiment

Autors: 
Kuznetsov Aleksandr Petrovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Popova Elena Sergeevna, Saratov State University
Seleznev Evgeny Petrovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Stankevich Nataliya Vladimirovna, National Research University "Higher School of Economics"
Abstract: 

In present paper nonlinear oscillator driving by external force in a form of three harmonic signals with irrational ratios of the frequencies and the map of various dynamical regimes on the parameter plane are presented. The feature of tri-frequencies torus doubling and destruction are investigated.

Key words: 
quasi-periodic oscillations
invariant tori
doubling of torus
Reference: 
  1. Pikovsky A, Rosenblum M, Kurts Yu. Synchronization. A fundamental nonlinear phenomenon. Moscow: Tehnosphera; 2003. 508 p. (In Russian).
  2. Loskutov AY, Mikhailov AS. Fundamentals of complex systems theory. Мoscow-Izhevsk: ICS; 2007. 620 p. (In Russian).
  3. Landa PS. Self-oscillations in systems with a finite number of degrees of freedom. Moscow: Nauka; 1980. 360 p. (In Russian).
  4. Anishchenko Sun, Astakhov VV, Vadivasova TE, Strelkova GI. Synchronization of regular, chaotic and stochastic fluctuations. Мoscow-Izhevsk: ICS; 2008. 144 p. (In Russian).
  5. Landau LD. To the problem of turbulence. Doklady Academy of Sciences USSR. 1944;44:339–342.
  6. Ruelle D, Takens F. On the nature of turbulence. Comm. Math. Phys. 1971;20(3):167–192.
  7. Vitolo R, Broer H, Simo C. Routes to chaos in the Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms. Nonlinearity. 2010;23(8):1919–1947. DOI: 10.1088/0951-7715/23/8/007.
  8. Vitolo R. Bifurcations of attractors in 3D dieomorphisms: A study in experimental mathematics. PhD thesis. 2003. http://dissertations.ub.rug.nl/faculties/science/2003/r.vitolo/?pLanguag...
  9. Grebogi C, Ott E, Pelikan S, Yorke JA. Strange attractors that are not chaotic. Physica D. 1984;13(1,2):261–268. DOI: 10.1016/0167-2789(84)90282-3.
  10. Kuznetsov AP, Stankevich NV. A simple autonomous quasiperiodic self-oscillator. Communications in Nonlinear Science and Numerical Simulation. 2010;15(6):1676–1681. DOI: 10.1016/J.CNSNS.2009.06.027.
  11. Kuznetsov SP, Pikovsky AU, Foidel U. Strange non-chaotic attractor. Nonlinear waves 2004. Ed. Gaponov-Grekhov AV, Neorakin VI. Nizhny Novgorod: IAP RAS; 2005. 544 p. (In Russian).
  12. Bezruchko BP, Kuznetsov SP, Pikovsky AS, Feudel U, Seleznev EP. On Dynamics of nonlinear systems under external quasi-periodic force near the terminal point of the torus-doubling bifurcation curve. Izvestiya VUZ. Applied Nonlinear Dynamics. 1997;5(6):3–19.
  13. Bezruchko BP, Kuznetsov SP, Seleznev EP. Experimental observation of dynamics near the torus-doubling terminal critical point. Phys. Rev. E. 2000;62(6):7828–7830. DOI: 10.1103/PhysRevE.62.7828.
  14. Seleznev EP, Zaharevich AM. Control parameter space of a nonlinear oscillator under quasiperiodic driving. Izvestiya VUZ. Applied Nonlinear Dynamics. 2009;17(6):17–35. DOI: 10.18500/0869-6632-2009-17-6-17-35.
  15. Anishchenko VS, Safonova MA, Feudel U, Kurths J. Bifurcation and transition to chaos through three-dimensional tori. Int. J. of Bifurcation and Chaos. 1994;4(3):595–607. DOI: 10.1142/S0218127494000423.
  16. Popova ES. Influence of fluctuations on evolution of three-dimensional torus in nonautonomous system. Izvestiya VUZ. Applied Nonlinear Dynamics. 2012;20(2):98–103. DOI: 10.18500/0869-6632-2012-20-2-98-103.
  17. Kim S, Ostlund S. Simultaneous rational approximations in the study of dynamical systems. Phys. Rev. A. 1986;34(4):3426–3434. DOI: 10.1103/physreva.34.3426.
  18. Linsay PS, Cumming AW. Three-frequency quasiperiodisity, phase locking and the onset of chaos. Physica D. 1989;40:196–217.
  19. Moon FC, Holmes WT. Double Poincare sections of a quasi-periodically forced, chaotic attractor. Physics Letters A. 1985;111(4):157–160. DOI: 10.1063/1.165815.
  20. Kuznetsov AP., Popova EU, Seleznev EP, Stankevich NV. A DIAGNOSTIC APPROACH FOR THE MULTIFREQUENCY TORUS. Vestnik SSTU. 2013;1(1):33–39.
  21. Anishchenko VS, Nikolaev SM. Generator of quasi-periodic oscillations featuring two-dimensional torus doubling bifurcations. Technical Physics Letters. 2005;31(10):853–855. DOI: 10.1134/1.2121837.
Received: 
23.04.2013
Accepted: 
16.07.2013
Published: 
31.12.2013
Short text (in English):
PDF icon a2013no5p031.pdf
(downloads: 61)
  • 1910 reads