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


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Kuznetsov A. P., Stankevich N. V. Dynamics of coupled generators of quasi-periodic oscillations with equilibrium state. Izvestiya VUZ. Applied Nonlinear Dynamics, 2018, vol. 26, iss. 2, pp. 41-58. DOI: 10.18500/0869-6632-2018-26-2-41-58

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Language: 
Russian
Article type: 
Article
UDC: 
517.9

Dynamics of coupled generators of quasi-periodic oscillations with equilibrium state

Autors: 
Kuznetsov Aleksandr 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: 

Subject of the study. Recently, the problems of synchronization of systems demonstrating quasi-periodic oscillations arouse interest. In particular, it can be generators of quasi-periodic oscillations that allow a radiophysical realization. In this paper we consider the dynamics of two coupled oscillators of quasi-periodic oscillations with a single equilibrium state. Novelty. The difference from the already studied case of coupled modified Anishchenko– Astakhov generators consists in engaging of two-parameter analysis and analysis in a much wider range of parameter changes, as well as a more dimensionless equation for an individual generator. Methods. The method of charts of Lyapunov exponents is used, which reveals areas of various types of dynamics, up to four-frequency oscillations. The bifurcation mechanisms of complete synchronization are investigated. Results. The possibility of synchronous quasiperiodicity is demonstrated, when the phases of the generators are locked, but the dynamics of the system is generally quasi-periodic. The possibility of the effect of «death of oscillations» arising due to the dissipative character of coupling is revealed. The possibility of the effect of broadband quasi-periodicity is demonstrated. Its peculiarity consists in the fact that twofrequency oscillations arise in a certain range of variation of the coupling parameter and a wide range of frequency mismatch. The bifurcation mechanisms of this effect are presented. It is shown that a certain degeneracy is characteristic for it, which is removed when nonidentity is introduced along the control parameters of individual generators. A bifurcation analysis is presented for this case. Two-parameter analysis allowed us to identify points of quasiperiodic bifurcations of codimension two QSNF (Quasi-periodic saddle-node fan) on the parameter plane, associated with the synchronization of multi-frequency tori. These points are the tips of the tongues of the two-frequency regimes, which have a threshold for the coupling coefficient. In their vicinity, three- and four-frequency quasi-periodic regimes are also observed. Discussion. Synchronization of quasi-periodic generators has a number of new moments that are established in two-parameter analysis in a wide range of parametric changes.

Reference: 
  1. Pikovsky A., Rosenblum M., Kurths J. Synchronization: a universal concept in nonlinear sciences. Cambridge, England: Cambridge university press, 2003. 423 p.
  2. Rabinovich M.I. Trubetskov D.I. Introduction to the theory of oscillations and waves. M.-Ijevsk: Regulyarnaya i haoticheskaya dinamika, 1999. 560 p. (in Russian).
  3. Anishchenko V.S., Astakhov V.V., Vadivasova T.E., Strelkova G.I. Synchronization of regular, chaotic and stochastic oscillations. M.; Izhevsk: Institute for Computer Research, 2008. 144 p. (in Russian).
  4. Kuznetsov A.P., Sataev I.R., Stankevich N.V. Tyuryukina L.V. Physics of quasiperiodic oscillations. Saratov: Publishing Center «Nauka», 2013. 252 p. (in Russian).
  5. Anishchenko V., Astakhov S., Vadivasova T. Phase dynamics of two coupled oscillators under external periodic force. Europhysics Letters, 2009, vol. 86, p. 30003.
  6. Anishchenko V.S., Astakhov S.V., Vadivasova T.E., Feoktistov A.V. Numerical and experimental investigation of external synchronization of two-frequency oscillations. Nelineinaya dinamika, 2009, vol. 5, no. 2, p. 237 (in Russian).
  7. Anishchenko V.S., Nikolaev S.M. Mechanisms of synchronization of a resonance limit cycle on a two-dimensional torus. Nelineinaya dinamika, 2008, vol. 4, no. 1, p. 39 (in Russian).
  8. Anishchenko V., Nikolaev S., Kurths J. Bifurcational mechanisms of synchronization of a resonant limit cycle on a two-dimensional torus. CHAOS, 2008, vol. 18, p. 037123.
  9. Anishchenko V.S., Nikolaev S.M., Kurths J. Peculiarities of synchronization of a resonant limit cycle on a two-dimensional torus. Phys. Rev. Е, 2007, vol.76, no. 4, p. 046216.
  10. Anishchenko V., Nikolaev S. Generator of quasi-periodic oscillations featuring twodimensional torus doubling bifurcations. Technical Physics Letters, 2005, vol. 31, no. 10, p. 853.
  11. Anishchenko V.S., Nikolaev S.M. Stability, synchronization and destruction of quasiperiodic motions. Nelineinaya dinamika, 2006, vol. 2, no. 3, p. 267 (in Russian).
  12. Anishchenko V., Nikolaev S., Kurths J. Winding number locking on a two-dimensional torus: Synchronization of quasiperiodic motions. Phys. Rev. E, 2006, vol. 73, no. 5, p. 056202.
  13. Broer H, Simo C., Vitolo R. Quasi-periodic bifurcations of invariant circles in low- ´ dimensional dissipative dynamical systems. Regular and Chaotic Dynamics, 2011, vol. 16, no. 1-2, p. 154. 
  14. Komuro M., Kamiyama K., Endo T., Aihara K. Quasi-periodic bifurcations of higher-dimensional tori. Int. J. of Bifurcation and Chaos, 2016, vol. 26, no. 7, p. 1630016.
  15. Broer H, Simo C., Vitolo R. The Hopf-saddle-node bifurcation for fixed points of ´ 3D-diffeomorphisms: the Arnol’d resonance web. Reprint from the Belgian Mathematical Society, 2008, p. 769.
  16. Stankevich N. V., Kurths J., Kuznetsov A. P. Forced synchronization of quasiperiodic oscillations. Communications in Nonlinear Science and Numerical Simulation, 2015, vol. 20, no. 1, p. 316.
  17. Rosenblum M., Pikovsky A. Self-organized quasiperiodicity in oscillator ensembles with global nonlinear coupling. Physical review letters, 2007, vol. 98, no. 6, p. 064101.
  18. Pikovsky A., Rosenblum M. Self-organized partially synchronous dynamics in populations of nonlinearly coupled oscillators. Physica D, 2009, vol. 238, no. 1, p. 27.
  19. Rosenblum M., Pikovsky A. Two types of quasiperiodic partial synchrony in oscillator ensembles. Phys. Rev. E, 2015, vol. 92, no. 1, p. 012919.
  20. Emelianova Yu. P., Kuznetsov A.P., Sataev I.R., Turukina L.V. Synchronization and multi-frequency oscillations in the low-dimensional chain of the self-oscillators. Physica D: Nonlinear Phenomena, 2013, vol. 244, no. 1, p. 36.
  21. Kuznetsov A.P., Kuznetsov S.P., Sataev I.R., Turukina L.V. About Landau–Hopf scenario in a system of coupled self-oscillators. Physics Letters A, 2013, vol. 377, p. 3291.
  22. Kuznetsov A.P. Migunova N.A., Sataev I.R., Sedova Yu.V., Turukina L.V. From chaos to quasi-periodicity. Regular and Chaotic Dynamics, 2015, vol. 20, no. 2, p. 189.
  23. Itoh K., Inaba N., Sekikawa M. Three-torus-causing mechanism in a third-order forced oscillator. Progress of Theoretical and Experimental Physics, 2013, no. 9, p. 093A02.
  24. Kamiyama K., Inaba N., Sekikawa M., Endo T. Bifurcation boundaries of threefrequency quasi-periodic oscillations in discrete-time dynamical system. Physica D, 2014, vol. 289, p. 12.
  25. Sekikawa M., Inaba N., Kamiyama K., Aihara K. Three-dimensional tori and Arnold tongues. Chaos, 2014, vol. 24, no. 1, p. 013137.
  26. Hidaka S., Inaba N., Sekikawa M., Endo T. Bifurcation analysis of four-frequency quasi-periodic oscillations in a three-coupled delayed logistic map. Physics Letters A, 2015, vol. 379, no. 7, p. 664.
  27. Kuznetsov A.P., Stankevich N.V. Synchronization of generators of quasiperiodic oscillations. Nelineinaya dinamika, 2013, vol. 9, no. 3, p. 409 (in Russian).
  28. Kuznetsov A.P., Kuznetsov S.P., Stankevich N. V. A simple autonomous quasiperiodic self-oscillator. Communications in Nonlinear Science and Numerical Simulation, 2010, vol. 15, no. 6, p. 1676. 
  29. Kuznetsov A.P., Kuznetsov S.P., Mosekilde E., Stankevich N.V. Generators of quasiperiodic oscillations with three-dimensional phase space. The European Physical Journal Special Topics, 2013, no.10, p. 2391.
  30. Kuznetsov A.P. Dynamic systems and bifurcations. Saratov: Publishing Center «Nauka», 2015. 168 p.
  31. Anishchenko V. S., Safonova M.A., Feudel U., Kurths J. Bifurcations and transition to chaos through three-dimensional tori. Int. J. of Bifurcation and Chaos, 1994, vol. 4, no. 03. P. 595. 
Received: 
10.03.2018
Accepted: 
10.04.2018
Published: 
26.04.2018
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