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

For citation:

Kuznetsov A. P., Stankevich N. V., Shchegoleva N. A. Synchronization of coupled generators of quasi-periodic oscillations upon destruction of invariant curve. Izvestiya VUZ. Applied Nonlinear Dynamics, 2021, vol. 29, iss. 1, pp. 136-159. DOI: 10.18500/0869-6632-2021-29-1-136-159

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: 1274)
Article type: 

Synchronization of coupled generators of quasi-periodic oscillations upon destruction of invariant curve

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"
Shchegoleva N. A., Yuri Gagarin State Technical University of Saratov

The purpose of this study is to describe the complete picture of synchronization of two coupled generators of quasi-periodic oscillations, to classify various types of synchronization, to study features of occurrence and destruction of multi-frequency quasi-periodic oscillations. Methods. The object of the research is systems of ordinary differential equations of various dimensions. The work uses the fourth-order Runge–Kutta method to solve a system of differential equations. Main analysis of the dynamics is carried out on the basis of calculated spectrum of Lyapunov exponents depending on parameters of systems, so-called charts of Lyapunov exponents. Bifurcation trees, winding numbers, phase portraits and Poincare maps were also visualized. ´ Results. Study of the dynamics of two coupled quasi-periodic generators for two sets of operating parameters of the subsystems is carried out. Two cases were studied when a two-frequency torus or chaotic oscillations (destroyed torus) are observed in the first oscillator. The dynamics of the second oscillator demonstrates different types of dynamics with a variation of the frequency detuning: periodic, quasi-periodic and chaotic. It is shown that for all parameters, phase synchronization of generators, broadband synchronization, and the phenomenon of oscillator death are observed. Dynamical regimes picture of the parameter plane frequency detuning – the coupling strength has a universal structure. Exit from the broadband synchronization region with a decrease in the coupling strength is accompanied by a quasiperiodic Hopf bifurcation and the birth of a three-frequency torus. Conclusion. The interaction of the simplest generators of quasi-periodic oscillations gives a rich picture of dynamic regimes: multi-frequency quasi-periodic oscillations with a different numbers of frequencies, chaotic behavior characterized by a different spectrum of Lyapunov exponents. Despite the variety of dynamic regimes, the synchronization picture of two dissipatively coupled quasi-periodic generators has a universal structure. Quasi-periodic phase and broadband synchronization are observed. The destruction of a torus in a subsystem leads to the destruction of multi-frequency tori in the system of two coupled oscillators, as well as a decrease in the variety of types of chaotic attractors.

Работа выполнена при поддержке гранта Президента Российской Федерации для поддержки молодых российских ученых – кандидатов наук (МК-31.2019.8). Работа А.П.К. выполнена в рамках государственного задания Института радиотехники и электроники им. В.А. Котельникова РАН. Работа Н.В.С. выполнена частично при поддержке Лаборатории динамических систем и приложений НИУ ВШЭ, грант Министерства науки и высшего образования РФ cоглашение № 075-15-2019-1931.
  1. Landa P.S. Nonlinear Oscillations and Waves in Dynamical Systems. Springer Science & Business Media, 1996, vol. 360, 544 p. DOI: 10.1007/978-94-015-8763-1.
  2. Anishchenko V.S., Vadivasova T.E., Strelkova G.I. Deterministic Nonlinear Systems. A Short Course: Springler Series in Synergetics. Springer International Publishing, Switzerland, 2014, 294 p. DOI: 10.1007/978-3-319-06871-8.
  3. Glazier J.A., Libchaber A. Quasi-periodicity and dynamical systems: An experimentalist’s view. IEEE Transactions on circuits and systems, 1988, vol. 35, no. 7, p. 790–809. DOI: 10.1109/31.1826.
  4. Kuznetsov A.P., Sataev I.R., Stankevich N.V., Turukina L.V. Physics of Quasiperiodic oscillations. Saratov: Publishing center «Science», 2013, 252 p. (in Russian).
  5. Anishchenko V.S., Nikolaev S.M. Generator of quasi-periodic oscillations featuring two-dimensional torus doubling bifurcations. Tech. Phys. Lett., 2005, vol. 31, no. 10, p. 853–855. DOI: 10.1134/1.2121837.
  6. Anishchenko V.S., Nikolaev S.M. Stability, synchronization and destruction of quasiperiodic motions. Russian Journal of Nonlinear Dynamics, 2006, vol. 2, no. 3, p. 267–278. DOI: 10.20537/nd0603001 (In Russian).
  7. Anishchenko V.S, Nikolaev S.M, Kurths J. Winding number locking on a two-dimensional torus: Synchronization of quasiperiodic motions. Phys. Rev. E, 2006, vol. 73, no. 5, p. 056202. DOI: 10.1103/PhysRevE.73.056202.
  8. 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. DOI: 10.1103/PhysRevE.76.046216.
  9. Anishchenko V.S., Nikolaev S.M., Kurths J. Synchronization mechanisms of resonant limit cycle on two-dimensional torus. Russian Journal of Nonlinear Dynamics, 2008, vol. 4, no. 1, p. 39–56. DOI: 10.20537/nd0801002 (In Russian).
  10. Anishchenko V.S., Nikolaev S.M., Kurths J. Bifurcational mechanisms of synchronization of a resonant limit cycle on a two-dimensional torus. CHAOS, 2008, vol. 18, no. 3, p. 037123. DOI: 10.1063/1.2949929.
  11. Anishchenko V.S., Astakhov S.V., Vadivasova T.E. Phase dynamics of two coupled oscillators under external periodic force. Europhysics Letters, 2009, vol. 86, no. 3, p. 30003. DOI: 10.1209/0295-5075/86/30003.
  12. Anishchenko V.S., Astakhov S.V., Vadivasova T.E., Feoktistov A.V. Numerical and experimaental study of external synchronization of two-frequency oscillations. Russian Journal of Nonlinear Dynamics, 2009, vol. 5, no. 2, p. 237–252. DOI: 10.20537/nd0902006 (In Russian).
  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. 154–184. DOI: 10.1134/S1560354711010060.
  14. Komuro M., Kamiyama K., Endo T., Aihara K. Quasi-periodic bifurcations of higher-dimensional tori. International Journal of Bifurcation and Chaos, 2016, vol. 26, no. 7, p. 1630016. DOI: 10.1142/S0218127416300160.
  15. Broer H., Simo C., Vitolo R. The Hopf-saddle-node bifurcation for fixed points of 3D-diffeomor- ´ phisms: the Arnol’d resonance web. Bulletin of the belgian mathematical society-Simon stevin, 2008, vol. 15, no. 5, p. 769–787. DOI: 10.36045/bbms/1228486406.
  16. Inaba N., Kamiyama K., Kousaka T., Endo T. Numerical and experimental observation of Arnol’d resonance webs in an electrical circuit, Physica D: Nonlinear Phenomena, 2015, vol. 311–312, no. 17, pp. 17–24. DOI: 10.1016/j.physd.2015.08.008.
  17. Truong T.Q., Tsubone T., Sekikawa M., Inaba N. Complicated quasiperiodic oscillations and chaos from driven piecewise-constant circuit: Chenciner bubbles do not necessarily occur via simple phase-locking. Physica D: Nonlinear Phenomena, 2017, vol. 341, no. 4, pp. 1–9. DOI: 10.1016/j.physd.2016.09.008.
  18. Truong T.Q., Tsubone T., Sekikawa M., Inaba N. Border-collision bifurcations and Arnol’d tongues in two coupled piecewise-constant oscillators. Physica D: Nonlinear Phenomena, 2020, vol. 401, no. 1, p. 132148. DOI: 10.1016/j.physd.2019.132148.
  19. Wieczorek S., Krauskopf B., Lenstra D. Mechanisms for multistability in a semiconductor laser with optical injection. Opt. Commun., 2000, vol. 183, no. 1–4, p. 215–226. DOI: 10.1016/S0030-4018(00)00867-1.
  20. Wieczorek S., Simpson T.B., Krauskopf B., Lenstra D. Bifurcation transitions in an optically injected diode laser: theory and experiment. Opt. Commun., 2003, vol. 215, no. 1–3, p. 125–134. DOI: 10.1016/S0030-4018(02)02191-0.
  21. Anchikov D.A., Shakirov A.P., Krents A.A., Molevich N.E., Pakhomov A.V. Multi-frequency tori in wide-aperture lasers. Phys. Wave Phen., 2016, vol. 24, no. 2, p. 108–113. DOI: 10.3103/S1541308X16020047.
  22. Wu M., Kalinikos B.A., Carr L.D., Patton C.E. Observation of spin-wave soliton fractals in magnetic film active feedback rings. Phys. Rev. Lett., 2006, vol. 96, no. 18, p. 187202. DOI: 10.1103/PhysRevLett.96.187202.
  23. Ustinov A.B., Demidov V.E., Kondrashov A.V., Kalinikos B.A., Demokritov S.O. Observation of the chaotic spin-wave soliton trains in magnetic films. Phys. Rev. Lett., 2011, vol. 106, no. 1, p. 017201. DOI: 10.1103/physrevlett.106.017201.
  24. Wang Z., Hagerstrom A., Anderson J.Q., Eykholt R., Tong W., Wu M., Carr L.D., Kalinikos B.A. Chaotic spin-wave solitons in magnetic film feedback rings. Phys. Rev. Lett., 2011, vol. 107, no. 11, p. 114102. DOI: 10.1103/PhysRevLett.107.114102.
  25. Kondrashov A.V., Ustinov A.B., Kalinikos B.A. Studying dynamic chaos in microwave ring generators based on normally magnetized ferromagnetic film. Tech. Phys. Lett., 2016, vol. 42, no. 2, p. 208–211. DOI: 10.1134/S1063785016020279.
  26. Emelianova Y.P., Emelyanov V.V., Ryskin N.M. Synchronization of two coupled multimode oscillators with time-delayed feedback. Communications in Nonlinear Science and Numerical Simulation, 2014, vol. 19, no. 10, p. 3778–3791. DOI: 10.1016/j.cnsns.2014.03.031.
  27. Emelyanov V.V., Emelianova Y.P., Ryskin N.M. The mutual synchronization of coupled delayed feedback klystron oscillators. Tech. Phys., 2016, vol. 61, no. 8, p. 1256–1261. DOI: 10.1134/S1063784216080089.
  28. Ju H., Neiman A.B., Shilnikov A.L. Bottom-up approach to torus bifurcation in neuron models. CHAOS, 2018, vol. 28, no. 10, p. 106317. DOI: 10.1063/1.5042078.
  29. Pikovsky A.S., Rosenblum M.G., Kurths J. Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge University Press, Cambridge, 2001. DOI: 10.1017/CBO9780511755743.
  30. Anishchenko V.S., Vadivasova T.E., Strelkova G.I. Synchronization of periodic self-sustained oscillations. Deterministic Nonlinear Systems, Springer Series in Synergetics, Springer, Cham, 2014, p. 217–243. DOI: 10.1007/978-3-319-06871-8_13.
  31. Balanov A., Janson N., Postnov D., Sosnovtseva O. Synchronization: From Simple to Complex. Springer Series in Synergetics. Springer-Verlag, Heidelberg, 2009, 426 p. DOI: 10.1007/978-3-540-72128-4.
  32. Hohl A., Gavrielides A., Erneux T., Kovanis V. Quasiperiodic synchronization for two delaycoupled semiconductor lasers. Phys. Rev. A, 1999, vol. 59, no. 5, p. 3941–3949. DOI: 10.1103/PhysRevA.59.3941.
  33. Mondal S., Pawar S.A., Sujith R.I. Synchronous behaviour of two interacting oscillatory systems undergoing quasiperiodic route to chaos. CHAOS, 2017, vol. 27, no. 10, p. 103119. DOI: 10.1063/1.4991744.
  34. Kuznetsov A.P., Kuznetsov S.P., Stankevich N.V. Autonomous generator of quasiperiodic oscillation. Izvestiya VUZ. Applied Nonlinear Dynamics, 2010, vol. 18, no. 2, p. 51–61. DOI: 10.18500/0869-6632-2018-26-2-41-58 (in Russian).
  35. Kuznetsov A.P., Kuznetsov S.P., Stankevich N.V. A simple autonomous quasiperiodic selfoscillator. Communications in Nonlinear Science and Numerical Simulation, 2010, vol. 15, no. 6, p. 1676–1681. DOI: 10.1016/j.cnsns.2009.06.027.
  36. Kuznetsov A.P., Kuznetsov S.P., Mosekilde E., Stankevich N.V. Co-existing hidden attractors in a radiophysical oscillator system. Journal of Physics A: Mathematical and Theoretical, 2015, vol. 48, no. 12, p. 125101. DOI: 10.1088/1751-8113/48/12/125101.
  37. Stankevich N.V., Volkov E.I. Multistability in a three-dimensional oscillator: tori, resonant cycles and chaos. Nonlinear Dynamics, 2018, vol. 94, no. 4, p. 2455–2467. DOI: 10.1007/s11071-018-4502-9.
  38. 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, vol. 222, no. 10, p. 2391–2398. DOI: 10.1140/epjst/e2013-02023-x.
  39. Gonchenko S.V., Kazakov A.O., Turaev D. Wild pseudohyperbolic attractors in a four-dimensional Lorenz system. arXiv preprint arXiv:1809.07250, 2018.
  40. Kuznetsov A.P., Stankevich N.V. Synchronization of generators of quasiperiodic oscillations. Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 3, p. 409–419. DOI: 10.20537/nd1303002 (In Russian).
  41. 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, no. 2, p. 41–58. DOI: 10.18500/0869-6632-2018-26-2-41-58 (In Russian).
  42. Kuznetsov A.P., Kuznetsov S.P., Shchegoleva N.A., Stankevich N.V. Dynamics of coupled generators of quasiperiodic oscillations: Different types of synchronization and other phenomena. Physica D: Nonlinear Phenomena, 2019, vol. 398, no. 12, p. 1–12. DOI: 10.1016/j.physd.2019.05.014.
  43. Stankevich N.V., Shchegoleva N.A., Sataev I.R., Kuznetsov A.P. Three-dimensional torus breakdown and chaos with two zero Lyapunov exponents in coupled radio-physical generators. Journal of Computational and Nonlinear Dynamics, 2020, vol. 15, no. 11, p. 111001. DOI: 10.1115/1.4048025.
  44. Xiao-Wen L., Zhi-Gang Z. Phase Synchronization of coupled Rossler oscillators: Amplitude ¨ effect. Communications in Theoretical Physics, 2007, vol. 47, no. 2, p. 265–269. DOI: 10.1088/0253-6102/47/2/016.
  45. Pazo D., S ´ anchez E., Mat ´ ´ıas M.A. Transition to high-dimensional chaos through quasiperiodic motion. International Journal of Bifurcation and Chaos, 2001, vol. 11, no. 10, p. 2683–2688. DOI: 10.1142/S0218127401003747.
  46. Pazo D., Mat ´ ´ıas M.A. Direct transition to high-dimensional chaos through a global bifurcation. Europhysics Letter, 2005, vol. 72, no. 2, p. 176–182. DOI: 10.1209/epl/i2005-10239-3.
  47. Osipov G.V., Pikovsky A.S., Rosenblum M.G., Kurths J. Phase synchronization effects in a lattice of nonidentical Rossler oscillators. ¨ Phys. Rev. E, 1997, vol. 55, no. 3, p. 2353–2361. DOI: 10.1103/PhysRevE.55.2353.
  48. Kuznetsov A.P., Migunova N.A., Sataev I.R., Sedova Y.V., Turukina L.V. From chaos to quasiperiodicity. Regular and Chaotic Dynamics, 2015, vol. 20, no. 2, p. 189–204. DOI: 10.1134/S1560354715020070.
  49. Benettin G., Galgani L., Giorgilli A., Strelcyn J.-M. Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them. Part 1: Theory. Meccanica, 1980, vol. 15, no. 1, p. 9–20. DOI: 10.1007/BF02128236.
  50. Afraimovich V.S., Shilnikov L.P. Invariant tori, their destruction and stochasticity. Methods of the qualitative theory of differential equations: Interuniversity thematic collection of scientific papers, E.A. Leontovich–Andronova. Gorky: GSU, 1983, p. 3–26 (in Russian).
  51. Aframovich V.S., Shilnikov L.P. Strange attractors and quasiattractors. Nonlinear Dynamics and Turbulence. Eds. G.I. Barenblatt, G. Iooss, D. D. Joseph. Boston: Pitmen, 1983, p. 1–34.
  52. Gonchenko A.S., Gonchenko S.V., Shilnikov L.P. Towards scenarios of chaos appearance in three-dimensional maps. Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 1, p. 3–28. DOI: 10.20537/nd1201001 (In Russian).
  53. Gonchenko A.S., Gonchenko S.V., Kazakov A.O., Turaev D. Simple scenarios of onset of chaos in three-dimensional maps. International Journal of Bifurcation and Chaos, 2014, vol. 24, no. 08, p. 1440005. DOI: 10.1142/S0218127414400057.
  54. Emelianova Y.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. 36–49. DOI: 10.1016/j.physd.2012.10.012.
  55. Zhusubaliyev Z.T., Mosekilde E. Novel routes to chaos through torus breakdown in non-invertible maps. Physica D: Nonlinear Phenomena, 2009, vol. 238, no. 5, p. 589–602. DOI: 10.1016/j.physd.2008.12.012.