Izvestiya of Saratov University.

Physics

ISSN 1817-3020 (Print)
ISSN 2542-193X (Online)

For citation:

Kuznetsov A. P., Sedova Y. V. Ensembles of four discrete phase oscillators. Izvestiya of Saratov University. Physics , 2025, vol. 25, iss. 2, pp. 134-146. DOI: 10.18500/1817-3020-2025-25-2-134-146, EDN: ZUYNTS

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
30.06.2025
Full text:
download
(downloads: 237)
Language: 
Russian
Heading: 
Radiophysics, electronics, acoustics
Article type: 
Article
UDC: 
530.182
DOI: 
10.18500/1817-3020-2025-25-2-134-146
EDN: 
ZUYNTS

Ensembles of four discrete phase oscillators

Autors: 
Kuznetsov Alexander Petrovich, Saratov Branch of Kotelnikov Institute of Radioengineering and Electronics of Russian Academy of Sciences
Sedova Yuliya Viktorovna, Saratov Branch of Kotelnikov Institute of Radioengineering and Electronics of Russian Academy of Sciences
Abstract: 

Background and Objectives: Ensembles of four discrete phase oscillators are considered. The aim of the research is to study and compare ensembles with different topologies of coupling: chains, rings and stars. Materials and Methods: We carry out the analysis using three-dimensional discrete maps for the relative phases of oscillators. We use the method of Lyapunov exponent chart, which identifies periodic regimes, quasi-periodic regimes with a different number of incommensurable frequencies and chaos. The various modes are illustrated using phase portraits. Results: We have found the regions of different regimes in the frequency detuning space of oscillators for different topologies of coupling. Resonances are indicated and illustrated both for pairs and for triples of synchronized oscillators, which corresponds to three- and two-frequency quasi-periodicity. We observe Arnold resonance web based on four frequency as well as on chaotic regimes. Conclusion: The ensemble of four discrete phase oscillators demonstrates a variety of quasi-periodic regimes with a different number of incommensurable frequencies, which are caused by possible resonances depending on the topology of coupling.

Key words: 
phase
Kuramoto model
discrete map
Lyapunov exponents
quasi-periodicity
Acknowledgments: 
The study was carried in the framework of the State Task of Kotelnikov Institute of Radioengineering and Electronics of Russian Academy of Sciences.
Reference: 
  1. Pikovsky A., Rosenblum M., Kurths J. Synchronization: A Universal Concept in Nonlinear Science. Cambridge University Press, 2001. 411 р. https://doi.org/10.1017/CBO9780511755743
  2. Balanov A., Janson N., Postnov D., Sosnovtseva O. Synchronization: From Simple to Complex. Springer, 2009. 425 p.
  3. Strogatz S. H. From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators. Physica D, 2000, vol. 143, iss. 1–4, pp. 1–20. https://doi.org/10.1016/S0167-2789(00)00094-4
  4. Acebrón J. A., Bonilla L. L., Pérez Vicente C. J., Ritort F., Spigler R. The Kuramoto model: A simple paradigm for synchronization phenomena. Rev. of Mod. Phys., 2005, vol. 77, iss. 1, pp. 137–185. https://doi.org/10.1103/RevModPhys.77.137
  5. Kuznetsov A. P., Sedova Y. V., Stankevich N. V. Discrete Rössler Oscillators: Maps and Their Ensembles. Int. J. of Bifur. and Chaos, 2023, vol. 33, no. 15, art. 2330037. https://doi.org/10.1142/S0218127423300379
  6. Biju A. E., Srikanth S., Manoj K., Pawar S. A., Sujith R. I. Dynamics of minimal networks of limit cycle oscillators. Nonlinear Dynamics, 2024, vol. 112, pp. 11329–11348. https://doi.org/10.1007/s11071-024-09641-5
  7. Arefev A. M., Grines E. A., Osipov G. V. Heteroclinic cycles and chaos in a system of four identical phase oscillators with global biharmonic coupling. Chaos, 2023, vol. 33, iss. 8, art. 083112. https://doi.org/10.1063/5.0156446
  8. Ashwin P., Burylko O. Weak chimeras in minimal networks of coupled phase oscillators. Chaos, 2015, vol. 25, iss. 1, art. 013106. https://doi.org/10.1063/1.4905197
  9. Guan Y., Moon K., Kim K. T., Li L. K. Chimera states in a can-annular combustion system. INTER-NOISE and NOISE-CON Congress and Conference Proceedings, 2023, vol. 265, iss. 4, pp. 3350–3357. https://doi.org/10.3397/IN_2022{_}0473
  10. Maistrenko V., Vasylenko A., Maistrenko Y., Mosekilde E. Phase chaos in the discrete Kuramoto model. Int. J. of Bifur. and Chaos, 2010, vol. 20, no. 6, pp. 1811–1823. https://doi.org/10.1142/S0218127410026861
  11. Maistrenko V., Vasylenko A., Maistrenko Y., Mosekilde E. Phase chaos and multistability in the discrete Kuramoto model. Nonlinear Oscillations, 2008, vol. 11, pp. 229–241. https://doi.org/10.1007/s11072-008-0026-4
  12. Kuznetsov A. P., Sedova Y. V. Low-dimensional discrete Kuramoto model: Hierarchy of multifrequency quasiperiodicity regimes. Int. J. of Bifur. and Chaos, 2014, vol. 24, no. 7, art. 1430022. https://doi.org/10.1142/S0218127414300225
  13. Shim W. On the generic complete synchronization of the discrete Kuramoto model. Kinetic & Related Models, 2020, vol. 13, iss. 5, pp. 979–1005. https://doi.org/10.3934/krm.2020034
  14. Broer H., Simó C., Vitolo R. The Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms: the Arnol’d resonance web. Bull. Belg. Math. Soc. Simon Stevin, 2008, vol. 15, iss. 5, pp. 769–787. https://doi.org/10.36045/bbms/1228486406
  15. Barlev G., Girvan M., Ott E. Map model for synchronization of systems of many coupled oscillators. Chaos, 2010, vol. 20, iss. 2, art. 023109. https://doi.org/10.1063/1.3357983
  16. Ha S. Y., Kim D., Kim J., Zhang X. Uniform-in-time transition from discrete to continuous dynamics in the Kuramoto synchronization. J. of Mathematical Phys., 2019, vol. 60, iss. 5, art. 051508. https://doi.org/10.1063/1.5051788
  17. Kim S., MacKay R. S., Guckenheimer J. Resonance regions for families of torus maps. Nonlinearity, 1989, vol. 2, no. 3, pp. 391–404. https://doi.org/10.1088/0951-7715/2/3/001
  18. Baesens С., Guckenheimer J., Kim S., MacKay R. S. Three coupled oscillators: mode-locking, global bifurcations and toroidal chaos. Physica D, 1991, vol. 49, iss. 3, pp. 387–475. https://doi.org/10.1016/0167-2789(91)90155-3
  19. Kuznetsov A. P., Sataev I. R., Sedova Y. V., Turukina L. V. On modelling the dynamics of coupled self-oscillators using the simplest phase maps. Izvestiya VUZ. Applied Nonlinear Dynamics, 2012, vol. 20, no. 2, pp. 112–137 (in Russian). https://doi.org/10.18500/0869-6632-2012-20-2-112-137
  20. Chen J., Zhou L., Sun W. Consensus analysis of chain star networks coupled by leaf nodes. Physica Scripta, 2023, vol. 98, no. 12, art. 125204. https://doi.org/10.1088/1402-4896/ad0588
  21. Chen X., Li F., Liu S., Zou W. Emergent behavior of conjugate-coupled Stuart–Landau oscillators in directed star networks. Physica A, 2023, vol. 629, art. 129211. https://doi.org/10.1016/j.physa.2023.129211
  22. Li X. Y., Chang J. M. LP-Star: Embedding Longest Paths into Star Networks with Large-Scale Missing Edges under an Emerging Assessment Model. IEEE TETC, 2025, vol. 13, pp. 147–161. https://doi.org/10.1109/TETC.2024.3387119
  23. Kuznetsov A. P., Sataev I. R., Turukina L. V. Regional Structure of Two-and Three-Frequency Regimes in a Model of Four Phase Oscillators. Int. J. of Bifur. and Chaos, 2022, vol. 32, no. 3, art. 2230008. https://doi.org/10.1142/S0218127422300087
  24. Emelianova Y. P., Kuznetsov A. P., Turukina L. V., Sataev I. R., Chernyshov N. Y. A structure of the oscillation frequencies parameter space for the system of dissipatively coupled oscillators. Commun. Nonlinear Sci. Numer. Simul., 2014, vol. 19, iss. 4, pp. 1203–1212. https://doi.org/10.1016/j.cnsns.2013.08.004
  25. Ashwin P., Guaschi J., Phelps J. M. Rotation sets and phase-locking in an electronic three oscillator system. Physica D, 1993, vol. 66, iss. 3–4, pp. 392–411. https://doi.org/10.1016/0167-2789(93)90075-C
  26. 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, no. 1, pp. 27–54 (in Russian). https://doi.org/10.18500/0869-6632-2014-22-1-27-54
Received: 
27.11.2024
Accepted: 
20.01.2025
Published: 
30.06.2025
  • 667 reads