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

For citation:

Kuznetsov A. P., Sataev I. R., Turukina L. V. Phase dynamics of periodically driven quasiperiodic self­-vibrating oscillators. Izvestiya VUZ. Applied Nonlinear Dynamics, 2010, vol. 18, iss. 4, pp. 17-32. DOI: 10.18500/0869-6632-2010-18-4-17-32

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):
PDF icon 2010no4p017.pdf
(downloads: 198)
Language: 
Russian
Heading: 
Applied Problems of Nonlinear Oscillation and Wave Theory
Article type: 
Article
UDC: 
517.9
DOI: 
10.18500/0869-6632-2010-18-4-17-32

Phase dynamics of periodically driven quasiperiodic self­-vibrating oscillators

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: 

Synchronization phenomena are studied in phase dynamics approximation in the periodically driven system of two coupled oscillators. The cases are discussed when the autonomous oscillators demonstrate phase locking or beats with incommensurate frequencies. Lyapunov charts are presented, the possible regimes of dynamics of the driven system are discussed. Different types of two-dimensional tori are revealed and classified. The modification of computer generated charts of dynamical regimes method is suggested to identify the domains of existence for different two-dimensional tori.

Key words: 
synchronization
phase oscillators
quasiperiodic dynamics
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. Anishchenko VS,  Astakhov SV,  Vadivasova TE, Strelkova GI. Synchronization of Regular, Chaotic, and Stochastic Oscillations. Moscow-Izhevsk: Institute of Computer Researches; 2008. 144 p. (in Russian).
  5. Landau LD. On the problem of turbulence. Dokl. Akad. Nauk SSSR. 1944;44(8):339–342 (in Russian) .
  6. Hopf E. A mathematical example displaying the features of turbulence. Communications on Pure and Applied Mathematics. 1948;1(4):303–322. DOI: 10.1002/cpa.3160010401.
  7. 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.
  8. 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).
  9. Anishchenko VS, Nikolaev SM. Synchronization of two-frequency quasi-periodic oscillations. Izvestiya VUZ. Applied Nonlinear Dynamics. 2008;16(2):69–86 (in Russian). DOI: 10.18500/0869-6632-2008-16-2-69-86.
  10. 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.
  11. 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.
  12. Scholarpedia. Phase model. http://www.scholarpedia.org/article/Phase_model/
  13. Kuznetsov SP. Dynamical Chaos: Course of Lectures. Moscow: Fizmatlit; 2001. 296 p. (in Russian).
  14. Van der Pol B. A theory of the amplitude of free and forced triode vibration. Radio Review. 1920;1:701–710.
  15. Bogoliyubov NN, Mitropolsky YuA. Asymptotic methods in the theory of nonlinear oscillations. Gostekhizdat; 1958. 406 p. (in Russian).
  16. Andronov AA, Witt AA, Haikin SE. Theory of oscillations. Vol. 2. Moscow: Fizmatlit; 1959. 916 p. (in Russian).
  17. Kuznetsov AP, Kuznetsov SP, Ryskin NM. Nonlinear oscillations. 2nd ed. Moscow: Fizmatlit; 2005. 292 p. (in Russian).
  18. Arnol'd VI, Ilyashenko YuS. Ordinary differential equations. In: Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Fund. Napr. Moscow: VINITI; 1985;1:7–140; Encyclopaedia Math. Sci. 1988;1:1–148.
  19. 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.
  20. Baesens C, 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.
Received: 
15.12.2009
Accepted: 
20.05.2010
Published: 
29.10.2010
Short text (in English):
PDF icon a2010no4p017.pdf
(downloads: 95)
  • 1702 reads