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

For citation:

Kuznetsov A. P., Sataev I. R., Turukina L. V. On the way towards multidimensional tori. Izvestiya VUZ. Applied Nonlinear Dynamics, 2010, vol. 18, iss. 6, pp. 65-84. DOI: 10.18500/0869-6632-2010-18-6-65-84

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 2010no6p065.pdf
(downloads: 149)
Language: 
Russian
Heading: 
Bifurcations in Dynamical Systems
Article type: 
Article
UDC: 
517.9
DOI: 
10.18500/0869-6632-2010-18-6-65-84

On the way towards multidimensional tori

Autors: 
Kuznetsov Aleksandr Petrovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Sataev Igor Rustamovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Turukina L. V., Saratov State University
Abstract: 

The problem of the dynamics of three coupled self-oscillators and three coupled periodically driven self-oscillators is discussed, in the last case only one of the oscillators is directly exited by the external fore. The regions of complete synchronization, two-, threeand four-frequency tori and chaos are revealed. Three typical situations of synchronization of three self-oscillators by the external driving are found. First situation refers to the mode locking of autonomous oscillators. Two other situations refer to quasi-periodic dynamics in the coupled autonomous oscillators. It is shown that multi-dimensional tori are not replaced by chaos and may dominate in the latter two cases. For the non-autonomous system under consideration the types of dynamics of three coupled oscillators are found, for which the complete synchronization of the system by the external driving is impossible independently of signal’s amplitude and frequency.

Key words: 
synchronization
phase oscillators
Quasi-periodic dynamics
chaos
Reference: 
  1. Pikovsky A, Rosenblum M, Kurts Yu. Synchronization: A fundamental nonlinear phenomenon. Moscow: Tehnosphera; 2003. 493 p.
  2. Landa PS. Self-Oscillation in Systems with Finite Number of Degress of Freedom. Moscow: Nauka; 1980. 360 p. (in Russian).
  3. Landa PS. Nonlinear Oscillations and Waves. Moscow: Nauka; 1997. 495 p. (in Russian). 
  4. Blekhman II. Synchronization in nature and technology. Moscow: Nauka; 1981. 351 p. (in Russian).
  5. Anishchenko VS, Astakhov SV, Vadivasova TE, Strelkova GI. Synchronization of Regular, Chaotic, and Stochastic Oscillations. Moscow-Izhevsk: Institute of Computer Investigations; 2008. (in Russian).
  6. Landau LD. On the problem of turbulence. Dokl. Akad. Nauk SSSR. 1944;44(8):339–342 (in Russian).
  7. Hopf E. A mathematical example displaying the features of turbulence. Communications on Pure and Applied Mathematics. 1948;1:303–322. DOI: 10.1002/cpa.3160010401.
  8. Ruelle D, Takens F. On the nature of turbulence. Commun. Math. Phys. 1971;20(3):167–192.
  9. Anishchenko V, Nikolaev S, Kurths J. Bifurcational mechanisms of synchronization of a resonant limit cycle on a two-dimensional torus. CHAOS. 2008;18:037123. DOI: 10.1063/1.2949929.
  10. Anishchenko VS, Nikolaev SM, Kurths J. Synchronization mechanisms of resonant limit cycle on two-dimensional torus. Nelin. Dinam. 2008;4(1):39–56 (in Russian). 
  11. Anishchenko VS, Nikolaev SM. Synchronization of two-frequency quasi-periodic oscillations. Izvestiya VUZ. Applied Nonlinear Dynamics. 2008;16(2):69–86. DOI: 10.18500/0869-6632-2008-16-2-69-86 (in Russian).
  12. Anishchenko V, Astakhov S, Vadivasova T. Phase dynamics of two coupled oscillators under external periodic force. Europhysics Letters. 2009;86(3):30003. DOI: http://dx.doi.org/10.1209/0295-5075/86/30003.
  13. Anishchenko VS, Astakhov SV, Vadivasova TE, Feoktistov AV. Numerical and experimental study of external synchronization of two-frequency oscillations. Nelin. Dinam. 2009;5(2):237–252.
  14. Kuznetsov AP, Sataev IR, Tyuryukina LV. Synchronization of quasi-periodic oscillations in coupled phase oscillators. Technical Physics Letters. 2010;36(5):478–481. DOI: 10.1134/S1063785010050263.
  15. Matrosov VV, Kasatkin DV. Particularities of dynamics for three cascade-coupled generators with phase control. Izvestiya VUZ. Applied Nonlinear Dynamics. 2004;12(1-2):159–168 (in Russian).
  16. Ponomarenko VP, Matrosov VV. Regimes of behaviour in the system of coupled oscillators with phase control. Izvestiya VUZ. Applied Nonlinear Dynamics. 2000;8(4):52–65 (in Russian).
  17. Matrosov VV, Korzinova MV. Cooperative dynamics of cascade coupling phase systems. Izvestiya VUZ. Applied Nonlinear Dynamics, 1994;2(2):10–16.
  18. Melnikova VA. PhD thesis «On synchronization of multimode generators»; 1977. (in Russian).
  19. Kuznetsov AP, Kuznetsov SP, Ryskin NM. Nonlinear oscillations. 2nd ed. Moscow: Fizmatlit; 2002. 292 p. (in Russian).
  20. Battelino PM. Persistence of three-frequency quasiperiodicity under large perturbations. Phys. Rev. A. 1988;38(3):1495–1502. DOI: 10.1103/physreva.38.1495.
  21. Pazo D, Sanchez E, Matias MA. Transition to high-dimensional chaos through quasiperiodic motion. Journal of Bifurcation and Chaos. 2001;11(10):2683–2688. DOI: 10.1142/S0218127401003747.
  22. Arnol'd VI, Ilyashenko YuS. Ordinary differential equations. Itogi Nauki i Tekhniki. Series: Sovrem. Probl. Mat. Encyclopaedia Math. Sci. 1988;1:1–148.
  23. Keith WL, Rand RH. 1:1 and 2:1 phase entrainment in a system of two coupled limit cycle oscillators. Journal of Mathematical Biology. 1984;20:133–152. DOI: 10.1007/BF00285342.
  24. Kuznetsov SP. Dynamical Chaos: Course of Lectures. Moscow: Fizmatlit; 2001. 296 p. (in Russian)
  25. Baesens С, Guckenheimer J, Kim S, MacKay RS. Three coupled oscillators: mode locking, global bifurcations and toroidal chaos. Physica D. 1991;49(3):387–475. DOI: 10.1016/0167-2789(91)90155-3.
  26. Linsay PS, Cumming AW. Three-frequency quasiperiodicity, phase locking, and the onset of chaos. Physica D. 1989;40:196–217. DOI: 10.1016/0167-2789(89)90063-8.
  27. Ashwin P, Burylko O, Maistrenko Y. Bifurcation to heteroclinic cycles and sensitivity in three and four coupled phase oscillators. Physica D. 2008;237(4):454–466. DOI: 10.1016/j.physd.2007.09.015.
  28. Karabacak O, Ashwin P. Heteroclinic Ratchets in Networks of Coupled Oscillators. J. Nonlinear Sci. 2010;20:105–129. DOI: 10.1007/s00332-009-9053-2.
Received: 
02.03.2010
Accepted: 
05.10.2010
Published: 
31.01.2011
Short text (in English):
PDF icon a2010no6p065.pdf
(downloads: 92)
  • 1925 reads