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


For citation:

Kuznetsov A. P., Turukina L. V., Sataev I. R., Chernyshov N. Y. Synchronization and multi-frequency quasi-periodicity in the dynamics of coupled oscillators. Izvestiya VUZ. Applied Nonlinear Dynamics, 2014, vol. 22, iss. 1, pp. 27-54. DOI: 10.18500/0869-6632-2014-22-1-27-54

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: 211)
Language: 
Russian
Article type: 
Article
UDC: 
517.9

Synchronization and multi-frequency quasi-periodicity in the dynamics of coupled oscillators

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

The dynamics of ensembles of oscillators containing a small number of bibitemlits is discussed. The possible types of regimes and pecularities of bifurcations of regular and quasi-periodic attractors are analyzed. By using the method of Lyapunov exponents charts the picture of  embedding of quasi-periodic regimes of different dimension in the parameter space is revealed. Dynamics of ensembles of van der Pol and phase oscillators are compared.

Reference: 
  1. Landa PS. Self-oscillations in systems with finite number of degrees of freedom. Moscow : Nauka; 1980. 360 p.
  2. Pikovsky A, Rosenblum M, Kurts Yu. Synchronization: A fundamental nonlinear phenomenon. Moscow:Tehnosphera; 2003. 493 p.
  3. Anishchenko VS, Astakhov VV, Vadivasova TE. Regular and chaotic oscillations. Synchronization and influence of fluctuations. Textbook-monograph. Dolgoprudny: Intelect publishing House; 2009. 312 p.
  4. Shilnikov LP, Shilnikov AL, Turaev DV, Chua LO. Methods of qualitative theory in nonlinear dynamics. Moscow-Izhevsk: Institute of Computer Science; 2003. 443 p.
  5. Shilnikov LP, Shilnikov AL, Turaev DV, Chua LO. Methods of qualitative theory in nonlinear dynamics. Part II. Moscow-Izhevsk: RCD; 2009. 548 p.
  6. Balanov AG, Janson NB, Postnov DE, Sosnovtseva O. Synchronization: From simple to complex. Springer; 2009. 2009437 p.
  7. Grebogi C, Ott E, James A, Yorkea J. Attractors on an N-torus: Quasiperiodicity versus chaos. Physica D. 1985;15(3):354—373.
  8. Battelino PM. Persistence of three-frequency quasiperiodicity under large perturbations. Phys. Rev. A. 1988;38:1495—1502.
  9. Linsay PS, Cumming AW. Three-frequency quasiperiodicity, phase locking, and the onset of chaos. Physica D. 1989;40:196—217.
  10. Kim S, MacKay RS, Guckenheimer J. Resonance regions for families of torus maps. Nonlinearity. 1989;2(3):391—404.
  11. Baesens С, Guckenheimer J, Kim S, MacKay RS. Simple resonance regions of torus diffeomorphisms. Patterns and Dynamics in Reactive Media (ed. R.Aris et al.). IMA Vol. in Maths. and its Applications. Springer. 1989;37:1—9.
  12. 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.
  13. Galkin OG. Phase-locking for maps of a torus: A computer assisted study. Chaos. 1993;3(1):73—82. DOI: 10.1063/1.165966.
  14. Ashwin P, Guasch J, Phelps JM. Rotation sets and phase-locking in an electronic three oscillator system. Physica D. 1993;66(3-4):392.
  15. Khibnik AI, Braimanc Y, Kennedyd TAB, Wiesenfeldd K. Phase model analysis of two lasers with injected field. Physica D. 1998;111(1-4):295—310. DOI: 10.1016/S0167-2789(97)80017-6.
  16. Guckenheimer J, Khibnik A. Torus maps from weak coupling of strong resonances. In book: Methods of Qualitative Theory of Differential Equations and Related Topics. American Mathematical Society; 2000. 205 p.
  17. Vasylenko A, Maistrenko Yu, Hasler M. Modeling phase synchronization in systems of two and three coupled oscillators. Nonlinear Oscillations. 2004;7(3):301—317. DOI: 10.1007/s11072-005-0014-x.
  18. Maistrenko Y, Popovych O, Burylko O. Mechanism of desynchronization in the finite-dimensional Kuramoto model. Phys. Rev. Lett. 2004;93(8):084102. DOI: 10.1103/PhysRevLett.93.084102.
  19. Anishchenko V, Astakhov S, Vadivasova T. Phase dynamics of two coupled oscillators under external periodic force. Europhysics Letters. 2009;86(3):30003. DOI: 10.1209/0295-5075/86/30003.
  20. Anishchenko VS, Astakhov VV, Vadivasova TE, Feoktistov AV. Numerical and experimental study of external synchronization of two-frequency oscillations. Nelin. Dinam. 2009;5(2):237—252.
  21. Anishchenko VS, Nikolaev SM, Kurths J. Synchronization mechanisms of resonant limit cycle on two-dimensional torus. Nelin. Dinam. 2008;4(1):39—56.
  22. Anishchenko V, Nikolaev S, Kurths J. Bifurcational mechanisms of synchronization of a resonant limit cycle on a two-dimensional torus. Chaos. 2008;18(3):037123. DOI: 10.1063/1.2949929.
  23. Anishchenko V, Nikolaev S, Kurths J. Winding number locking on a two-dimensional torus: Synchronization of quasiperiodic motions. Phys. Rev. E. 2006;73:056202. DOI: 10.1103/PHYSREVE.73.056202.
  24. Kuznetsov AP, Stankevich NV. Synchronization of generators of quasiperiodic oscillations. Nelin. Dinam. 2013;9(3):409—419.
  25. Rompala K, Rand R, Howland H. Dynamics of three coupled van der Pol oscillators with application to circadian rhythms. Communications in Nonlinear Science and Numerical Simulation. 2007;12(5):794—803. DOI: 10.1016/J.CNSNS.2005.08.002.
  26. Broer Н, Simo С, Vitolo R. The Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms: The Arnol’d resonance web. Bulletin of the Belgian Mathematical Society – Simon Stevin. 2008;15(5):769—787. DOI: 10.36045/bbms/1228486406.
  27. Broer H, Simo C, Vitolo R. Quasi-periodic bifurcations of invariant circles in low-dimensional dissipative dynamical systems. Regular and Chaotic Dynamics. 2011;16(1-2):154—184. DOI: 10.1134/S1560354711010060.
  28. Broer H, Simo C, Vitolo R. Routes to chaos in the Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms. Nonlinearity. 2010;23(8):1919. DOI: 10.1088/0951-7715/23/8/007.
  29. Astakhov S, Fujiwara N, Gulay A, Tsukamoto N, Kurths J. Hopf bifurcation and multistability in a system of phase oscillators. Phys. Rev. E. 2013;88(3):032908. DOI: 10.1103/PhysRevE.88.032908.
  30. Kuznetsov AP, Sataev IR, Turukina LV. Forced synchronization of two coupled van der Pol self-oscillators. Nelin. Dinam. 2011;7(3):411–425.
  31. Kuznetsov AP, Sataev IR, Turukina LV. Synchronization of forced quasi-periodic coupled oscillators. Preprint nlin; 2011. arXiv: 1106.5382.
  32. Turukina LV, Chernyshov NJ. Synchronization of reactively coupled phase oscillators driven by external force. Izvestiya VUZ. Applied Nonlinear Dynamics. 2012;20(1):81—90. DOI: 10.18500/0869-6632-2012-20-1-81-90.
  33. Kuznetsov AP, Sataev IR, Turukina LV. On the road towards multidimensional tori. Communications in Nonlinear Science and Numerical Simulation. 2011;16(6):2371—2376. DOI: 10.1016/j.cnsns.2010.09.026.
  34. Kuznetsov AP, Sataev IR, Turukina LV. Synchronization and multi-frequency oscillations in the chain of phase oscillators. Nelin. Dinam. 2010;6(4):693—717.
  35. Emelianova YuP, Kuznetsov AP, Turukina LV. Dynamics of three coupled van der Pol oscillators with non-identical controlling parameters. Izvestiya VUZ. Applied Nonlinear Dynamics. 2011;19(5):76—90. DOI: 10.18500/0869-6632-2011-19-5-76-90.
  36. Kuznetsov AP, Turukina LV, Chernyschov NYu. Dynamics and synchronization of the three coupled oscillators with reactive type of coupling. Nelin. Dinam. 2013;9(1):11—25.
  37. Kuznetsov AP, Kuznetsov SP, Turukina LV, Sataev IR. Landau–Hopf scenario in the ensemble of interacting oscillators. Nelin. Dinam. 2012;8(5):863—873.
  38. Emelianova YuP, Kuznetsov AP, Sataev IR, Turukina LV. Synchronization and multi-frequency oscillations in the low-dimensional chain of the self-oscillators. Physica D. 2012;244(1):36—49. DOI: 10.1016/j.physd.2012.10.012.
  39. Kuznetsov AP, Kuznetsov SP, Sataev IR, Turukina LV. About Landau–Hopf scenario in a system of coupled self-oscillators. Physics Letters A. 2013;377(45-48):3291—3295. DOI: 10.1016/j.physleta.2013.10.013.
  40. Emelianova YP, Kuznetsov AP, Turukina LV. Quasi-periodic bifurcations and «amplitude death» in low-dimensional ensemble of van der Pol oscillators. Physics Letters A. 2014;378(3):153—157. DOI:10.1016/j.physleta.2013.10.049.
  41. Emelianova YP, Kuznetsov AP, Turukina LV, Sataev IR, Chernyshov NYu. A structure of the oscillation frequencies parameter space for the system of dissipatively coupled oscillators. Commun. Nonlinear Sci. Numer. Simul. 2014;19(4):1203—1212. DOI: 10.1016/j.cnsns.2013.08.004.
  42. Rand R, Holmes PJ. Bifurcation of periodic motions in two weakly coupled van der Pol oscillators. Int. J. Non-Linear Mechanics. 1980;15(4-5):387—399. DOI: 10.1016/0020-7462(80)90024-4.
  43. Ivanchenko M, Osipov G, Shalfeev V, Kurths J. Synchronization of two non-scalarcoupled limit-cycle oscillators. Physica D. 2004;189(1–2):8—30. DOI: 10.1016/J.PHYSD.2003.09.035.
  44. Kuznetsov AP, Stankevich NV, Turukina LV. Coupled van der Pol–Duffing oscillators: Phase dynamics and structure of synchronization tongues. Physica D. 2009; 238(14):1203—1215. DOI: 10.1016/j.physd.2009.04.001.
  45. Kryukov AK, Osipov GV, Polovinkin AV, Kurths J. Synchronous regimes in ensembles of coupled Bonhoeffer–van der Pol oscillators. Phys. Rev. E. 2009;79:046209. DOI: 10.1103/PhysRevE.79.046209.
  46. Kuznetsov AP, Roman JuP. Properties of synchronization in the systems of non-identical coupled van der Pol and van der Pol–Duffing oscillators. Broadband synchronization. Physica D. 2009;238(16):1499—1506. DOI: 10.1016/j.physd.2009.04.016.
  47. Landau LD. About the problem of turbulence. DAN SSSR. 1944;44(8):339—342. 
  48. 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.
  49. Privezentsev AP, Sablin NI, Filipenko NM, Fomenko GP. Nonlinear oscillations of virtual cathode in triode system. Soviet journal of communications technology & electronics. 1992;37:37—42.
  50. Magda II, Pashchenko AV, Romanov SS. Theory of feedback in generators with virtual cathode. Problems of atomic science and technology. Series: Plasma electronics and new methods of acceleration. 2003;4:167—170.
  51. Sze H, Price D, Harteneck B. Phase locking of two strongly coupled vircators. J. Appl. Phys. 1990;67(5):2278—2282.
  52. Repin BG, Dubinov AE. Phasing of three vircators simulated in terms of coupled van der pol oscillators. Technical Physics. 2006;51(4):489—494. DOI: 10.1134/S1063784206040153.
  53. Pampaloni E, Lapucci A. Locking-range analysis for three coupled lasers. Opt. Lett. 1993;18(22):1881—1883. DOI: 10.1364/ol.18.001881.
  54. Braimanc Y, Kennedy TA, Wiesenfeldd K, Khibnik AI. Entrainment of solid-state laser arrays. Phys. Rev. A. 1995;52(2):1500—1506. DOI: 10.1103/physreva.52.1500.
  55. Khibnik AI, Braimanc Y, Protopopescu V, Kennedy TA, Wiesenfeldd K. Amplitude dropout in coupled lasers. Phys. Rev. A. 2000;62:063815.
  56. Glova AF, Lysikov AYu. Phase locking of three lasers optically coupled with a spatial filter. Kvantovaya Elektronika. 2002;32(4):315—318.
  57. Glova AF. Phase locking of optically coupled lasers. Kvantovaya Elektronika. 2003;33(4):283—306.
  58. Vladimirov AG. Nonlinear dynamics and bifurcations in multimode and spatially distributed laser systems. Dissertation for the degree of Doctor of Physical and Mathematical Sciences; 2006.
  59. Lee TE, Cross MC. Pattern formation with trapped ions. Phys. Rev. Lett. 2011;106(14):143001. DOI: 10.1103/PhysRevLett.106.143001.
  60. Lee TE, Sadeghpour HR. Quantum simulation of quantum van der Pol oscillators with trapped ions. Preprint arXiv; 2013.
  61. Valkering TP, Hooijer CLA, Kroon MF. Dynamics of two capacitively coupled Josephson junctions in the overdamped limit. Physica D. 2000;135(1-2):137—153. DOI: 10.1016/S0167-2789(99)00116-5.
  62. Saitoh K, Nishino T. Phase locking in a double junction of Josephson weak links. Phys. Rev. B. 1991;44:7070.
  63. Czolczynski K, Perlikowski P, Stefanski A, Kapitaniak T. Why two clocks synchronize:Energy balance of the synchronized clocks. Chaos. 2011;21(2):023129. DOI: 10.1063/1.3602225.
  64. Kapitaniak M, Czolczynski K, Perlikowski P, Stefanski A, Kapitaniak T. Synchronization of clocks. Physics Reports. 2012;517(1–2):1—69. DOI: 10.1016/j.physrep.2012.03.002.
  65. Czolczynski K, Perlikowski P, Stefanski A, Kapitaniak T. Clustering of Huygens’ clocks. Prog. Theor. Phys. 2009;122(4):1027—1033. DOI: 10.1143/PTP.122.1027.
  66. Czolczynski K, Perlikowski P, Stefanski A, Kapitaniak T. Synchronization of theself-excited pendula suspended on the vertically displacing beam. Communications in Nonlinear Science and Numerical Simulation. 2013;18(2):386—400. DOI: 10.1016/j.cnsns.2012.07.007.
  67. Hong H, Strogatz SH. Kuramoto model of coupled ocillators with positive and negative coupling parameters: An example of conformist and contrarian oscillators. Phys. Rev. Lett. 2011;106(5):054102. DOI: 10.1103/PhysRevLett.106.054102.
Received: 
19.12.2013
Accepted: 
19.12.2013
Published: 
30.04.2014
Short text (in English):
(downloads: 76)