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

For citation:

Koronovskii A. A., Kurovskaya M. K., Moskalenko O. I. On the typicity of the explosive synchronization phenomenon in oscillator networks with the link topology of the “ring” and “small world” types. Izvestiya VUZ. Applied Nonlinear Dynamics, 2023, vol. 31, iss. 1, pp. 32-44. DOI: 10.18500/0869-6632-003027, EDN: ABUBJC

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):
Full text PDF(En):
Article type: 

On the typicity of the explosive synchronization phenomenon in oscillator networks with the link topology of the “ring” and “small world” types

Koronovskii Aleksei Aleksandrovich, Saratov State University
Kurovskaya Maria Konstantinovna, Saratov State University
Moskalenko Olga Igorevna, Saratov State University

Purpose of this study is to investigate the problem of how typical (or, conversely, unique) is the phenomenon of explosive synchronization in networks of nonlinear oscillators with topologies of links such as “ring” and “small world”, and, in turn, how the partial frequencies of the interacting oscillators must correlate with each other for the phenomenon of explosive synchronization in these networks can be possible. Methods. In this paper, we use an analytical description of the synchronous behavior of networks of nonlinear elements with “ring” and “small world” link topologies. To confirm the obtained results the numerical simulation is used. Results. It is shown that in networks of nonlinear oscillators with topologies of links such as “ring” and “small world”, the phenomenon of explosive synchronization can be observed for the different distributions of partial frequencies of network oscillators. Conclusion. The paper considers an analytical description of the behavior of network oscillators with “ring” and “small world” topologies of links and shows that the phenomenon of explosive synchronization in such networks is atypical, but not unique.

This work was supported by Russian Science Foundation, project No. 19-12-00037
  1. Boccaletti S, Latora V, Moreno V, Chavez M, Hwang DU. Complex networks: Structure and dynamics. Physics Reports. 2006;424(4–5):175–308. DOI: 10.1016/j.physrep.2005.10.009.
  2. Arenas A, Diaz-Guilera A, Kurths J, Moreno Y, Zhou C. Synchronization in complex networks. Physics Reports. 2008;469(3):93–153. DOI: 10.1016/j.physrep.2008.09.002.
  3. Boccaletti S, Almendral JA, Guan S, Leyva I, Liu Z, Sendina-Nadal I, Wang Z, Zou Y. Explosive transitions in complex networks’ structure and dynamics: Percolation and synchronization. Physics Reports. 2016;660:1–94. DOI: 10.1016/j.physrep.2016.10.004.
  4. Leyva I, Sevilla-Escoboza R, Buldu JM, Sendina-Nadal I, Gomez-Gardenes J, Arenas A, Moreno Y, Gomez S, Jaimes-Reategui R, Boccaletti S. Explosive first-order transition to synchrony in networked chaotic oscillators. Phys. Rev. Lett. 2012;108(16):168702. DOI: 10.1103/PhysRevLett. 108.168702.
  5. Leyva I, Navas A, Sendina-Nadal I, Almendral JA, Buldu JM, Zanin M, Papo D, Boccaletti S. Explosive transitions to synchronization in networks of phase oscillators. Scientific Reports. 2013;3:1281. DOI: 10.1038/srep01281.
  6. Gomez-Gardenes J, Gomez S, Arenas A, Moreno Y. Explosive synchronization transitions in scale-free networks. Phys. Rev. Lett. 2011;106(12):128701. DOI: 10.1103/PhysRevLett.106.128701.
  7. Pikovsky A, Rosenblum M, Kurths J. Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge: Cambridge University Press; 2001. 411 p. DOI: 10.1017/CBO9780511755743.
  8. Anishchenko VS, Vadivasova TE. Relationship between frequency and phase characteristics of chaos: Two criteria of synchronization. Journal of Communications Technology and Electronics. 2004;49(1):69–75.
  9. Pazo D. Thermodynamic limit of the first-order phase transition in the Kuramoto model. Phys. Rev. E. 2005;72(4):046211. DOI: 10.1103/PhysRevE.72.046211.
  10. Koronovskii AA, Kurovskaya MK, Moskalenko OI, Hramov A, Boccaletti S. Self-similarity in explosive synchronization of complex networks. Phys. Rev. E. 2017;96(6):062312. DOI: 10.1103/ PhysRevE.96.062312.
  11. Peron TKDM, Rodrigues FA. Determination of the critical coupling of explosive synchronization transitions in scale-free networks by mean-field approximations. Phys. Rev. E. 2012;86(5):056108. DOI: 10.1103/PhysRevE.86.056108.
  12. Zou Y, Pereira T, Small M, Liu Z, Kurths J. Basin of attraction determines hysteresis in explosive synchronization. Phys. Rev. Lett. 2014;112(11):114102. DOI: 10.1103/PhysRevLett.112.114102.
  13. Koronovskii AA, Kurovskaya MK, Moskalenko OI. On the possibility of explosive synchronization in small world networks. Izvestiya VUZ. Applied Nonlinear Dynamics. 2021;29(4):467–479 (in Russian). DOI: 10.18500/0869-6632-2021-29-4-467-479.
  14. Zhu L, Tian L, Shi D. Criterion for the emergence of explosive synchronization transitions in networks of phase oscillators. Phys. Rev. E. 2013;88(4):042921. DOI: 10.1103/PhysRevE.88.042921.
  15. Peron TKDM, Rodrigues FA. Explosive synchronization enhanced by time-delayed coupling. Phys. Rev. E. 2012;86(1):016102. DOI: 10.1103/PhysRevE.86.016102.
  16. Leyva I, Sendina-Nadal I, Almendral JA, Navas A, Olmi S, Boccaletti S. Explosive synchronization in weighted complex networks. Phys. Rev. E. 2013;88(4):042808. DOI: 10.1103/PhysRevE. 88.042808.
  17. Jiang X, Li M, Zheng Z, Ma Y, Ma L. Effect of externality in multiplex networks on one-layer synchronization. Journal of the Korean Physical Society. 2015;66(11):1777–1782. DOI: 10.3938/ jkps.66.1777.
  18. Su G, Ruan Z, Guan S, Liu Z. Explosive synchronization on co-evolving networks. EPL (Europhysics Letters). 2013;103(4):48004. DOI: 10.1209/0295-5075/103/48004.
  19. Hu X, Boccaletti S, Huang W, Zhang X, Liu Z, Guan S, Lai CH. Exact solution for first-order synchronization transition in a generalized Kuramoto model. Scientific Reports. 2014;4(1):7262. DOI: 10.1038/srep07262.
  20. Kuramoto Y. Self-entrainment of a population of coupled non-linear oscillators. In: Araki H, editor. International Symposium on Mathematical Problems in Theoretical Physics. Vol. 39 of Lecture Notes in Physics. Berlin, Heidelberg: Springer; 1975. P. 420–422. DOI: 10.1007/BFb0013365.
  21. Acebron JA, Bonilla LL, Perez-Vicente CJ, Ritort F, Spigler R. The Kuramoto model: A simple paradigm for synchronization phenomena. Rev. Mod. Phys. 2005;77(1):137–185. DOI: 10.1103/ RevModPhys.77.137.
  22. Watts DJ, Strogatz SH. Collective dynamics of ‘small-world’ networks. Nature. 1998;393(6684): 440–442. DOI: 10.1038/30918. 
Available online: