Abstract
Synchronization in complex networks is influenced by higher-order interactions and non-Gaussian perturbations, yet their mechanisms remain unclear. We investigate the synchronization and spike dynamics in a higher-order Kuramoto model subjected to Lévy noise. Using the mean order parameter, mean first-passage time, and basin stability, we identify boundaries distinguishing synchronization and incoherence. The stability index governs the tail heaviness of the probability density function for Lévy noise, while the scale parameter affects the magnitude. Synchronization weakens as the stability index decreases, and even completely disappears when the scale parameter exceeds a critical threshold. By varying coupling, we find bifurcations and hysteresis. Lévy noise smooths the synchronization transitions and requires stronger coupling compared to Gaussian white noise. We then define spikes as extreme excursions of the order parameter and study their statistical and spectral properties. The maximum number of spikes is observed at small-scale parameters. A generalized spectral analysis based on an edit distance algorithm measures the similarity between spike sequences and identifies spike patterns. These findings deepen the understanding of synchronization and extreme events in complex networks driven by non-Gaussian noise.
Similar content being viewed by others
Data availability
All data needed to evaluate the findings of the paper are available within the paper.
Code availability
The Python and Julia source code is available at https://github.com/zhaodan-npu/Higher_order_kuramoto_paper_code.
References
Shahal, S. et al. Synchronization of complex human networks. Nat. Commun. 11, 3854 (2020).
Glass, L. Synchronization and rhythmic processes in physiology. Nature 410, 277–284 (2001).
Blasius, B., Huppert, A. & Stone, L. Complex dynamics and phase synchronization in spatially extended ecological systems. Nature 399, 354–359 (1999).
Rohden, M., Sorge, A., Timme, M. & Witthaut, D. Self-organized synchronization in decentralized power grids. Phys. Rev. Lett. 109, 064101 (2012).
Arenas, A., Díaz-Guilera, A., Kurths, J., Moreno, Y. & Zhou, C. Synchronization in complex networks. Phys. Rep. 469, 93–153 (2008).
Gallo, L. et al. Synchronization induced by directed higher-order interactions. Commun. Phys. 5, 263 (2022).
Acebrón, J. A., Bonilla, L. L., Pérez Vicente, C. J., Ritort, F. & Spigler, R. The Kuramoto model: A simple paradigm for synchronization phenomena. Rev. Mod. Phys. 77, 137–185 (2005).
Rodrigues, F. A., Peron, T. K. D., Ji, P. & Kurths, J. The Kuramoto model in complex networks. Phys. Rep. 610, 1–98 (2016).
Strogatz, S. H. From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators. Phys. D. 143, 1–20 (2000).
Dorfler, F. & Bullo, F. Synchronization and transient stability in power networks and nonuniform Kuramoto oscillators. SIAM J. Control Optim. 50, 1616–1642 (2012).
Bick, C., Gross, E., Harrington, H. A. & Schaub, M. T. What are higher-order networks? SIAM Rev. 65, 686–731 (2023).
Benson, A. R., Gleich, D. F. & Leskovec, J. Higher-order organization of complex networks. Science 353, 163–166 (2016).
Skardal, P. S. & Arenas, A. Higher order interactions in complex networks of phase oscillators promote abrupt synchronization switching. Commun. Phys. 3, 218 (2020).
Skardal, P. S. & Arenas, A. Abrupt desynchronization and extensive multistability in globally coupled oscillator simplexes. Phys. Rev. Lett. 122, 248301 (2019).
Ferraz de Arruda, G., Tizzani, M. & Moreno, Y. Phase transitions and stability of dynamical processes on hypergraphs. Commun. Phys. 4, 24 (2021).
Millán, A. P., Torres, J. J. & Bianconi, G. Explosive higher-order Kuramoto dynamics on simplicial complexes. Phys. Rev. Lett. 124, 218301 (2020).
Xu, C. & Skardal, P. S. Spectrum of extensive multiclusters in the Kuramoto model with higher-order interactions. Phys. Rev. Res. 3, 013013 (2021).
Kundu, S. & Ghosh, D. Higher-order interactions promote chimera states. Phys. Rev. E 105, L042202 (2022).
Millán, A. P., Sun, H., Torres, J. J. & Bianconi, G. Triadic percolation induces dynamical topological patterns in higher-order networks. PNAS Nexus 3, 270 (2024).
Zhang, Y., Skardal, P. S., Battiston, F., Petri, G. & Lucas, M. Deeper but smaller: Higher-order interactions increase linear stability but shrink basins. Sci. Adv. 10, eado8049 (2024).
Albert, R., Albert, I. & Nakarado, G. L. Structural vulnerability of the North American power grid. Phys. Rev. E 69, 025103 (2004).
Wang, S., Gu, X., Luan, S. & Zhao, M. Resilience analysis of interdependent critical infrastructure systems considering deep learning and network theory. Int. J. Crit. Infrastruct. Prot. 35, 100459 (2021).
Bian, J., Zhou, T. & Bi, Y. Unveiling the role of higher-order interactions via stepwise reduction. Commun. Phys. 8, 228 (2025).
Goldbeter, A. Biological rhythms: clocks for all times. Curr. Biol. 18, R751–R753 (2008).
Dong, Y., Huo, L., Perc, M. & Boccaletti, S. Adaptive rumor propagation and activity contagion in higher-order networks. Commun. Phys. 8, 261 (2025).
Zhao, D., Li, Y., Liu, Q., Zhang, H. & Xu, Y. The occurrence mechanisms of extreme events in a class of nonlinear Duffing-type systems under random excitations. Chaos 33, 083109 (2023).
Wang, Z., Li, Y., Xu, Y., Kapitaniak, T. & Kurths, J. Coherence-resonance chimeras in coupled hr neurons with alpha-stable lévy noise. J. Stat. Mech. 2022, 053501 (2022).
Zhang, X., Xu, Y., Liu, Q., Kurths, J. & Grebogi, C. Rate-dependent bifurcation dodging in a thermoacoustic system driven by colored noise. Nonlinear Dyn. 104, 2733–2743 (2021).
Wang, Z., Xu, Y. & Yang, H. Lévy noise-induced stochastic resonance in an fhn model. Sci. China Technol. Sci. 59, 371–375 (2016).
Campa, A. & Gupta, S. Synchronization in a system of Kuramoto oscillators with distributed Gaussian noise. Phys. Rev. E 108, 064124 (2023).
Holder, A. B., Zuparic, M. L. & Kalloniatis, A. C. Gaussian noise and the two-network frustrated Kuramoto model. Phys. D. 341, 10–32 (2017).
Zhao, D., Li, Y., Xu, Y., Liu, Q. & Kurths, J. Probabilistic description of extreme oscillations and reliability analysis in rolling motion under stochastic excitation. Sci. China Technol. Sci. 66, 2586–2596 (2023).
Tyulkina, I. V., Goldobin, D. S., Klimenko, L. S. & Pikovsky, A. Dynamics of noisy oscillator populations beyond the Ott-Antonsen ansatz. Phys. Rev. Lett. 120, 264101 (2018).
Rajwani, P. & Jalan, S. Stochastic Kuramoto oscillators with inertia and higher-order interactions. Phys. Rev. E 111, L012202 (2025).
Marui, Y. & Kori, H. Synchronization and its slow decay in noisy oscillators with simplicial interactions. Phys. Rev. E 111, 014223 (2025).
Zhao, D., Li, Y., Liu, Q., Kurths, J. & Xu, Y. Extreme events suppression in a suspended aircraft seat system under extreme environment. Commun. Nonlinear Sci. Numer. Simul. 145, 108707 (2025).
Liu, Q., Xu, Y. & Li, Y. Complex dynamics of a conceptual airfoil structure with consideration of extreme flight conditions. Nonlinear Dyn. 111, 14991–15010 (2023).
Lee, S., Kuklinski, L. J. & Timme, M. Extreme synchronization transitions. Nature Communications 16, 4505 (2025).
Suman, A. & Jalan, S. Finite-size effect in Kuramoto oscillators with higher-order interactions. Chaos 34, 101101 (2024).
Zan, W., Jia, W. & Xu, Y. Response statistics of single-degree-of-freedom systems with lévy noise by improved path integral method. Int. J. Appl. Mech. 14, 2250029 (2022).
Zhao, D., Li, Y., Xu, Y., Liu, Q. & Kurths, J. Extreme events in a class of nonlinear Duffing-type oscillators with a parametric periodic force. Eur. Phys. J. 137,314 (2022).
Banerjee, A. et al. Recurrence analysis of extreme event-like data. Nonlinear Process. Geophys. Discuss. 2020, 1–25 (2020).
Marwan, N. & Braun, T. Power spectral estimate for discrete data. Chaos 33, 053118 (2023).
Marwan, N. Challenges and perspectives in recurrence analyses of event time series. Front. Appl. Math. Stat. 9, 1129105 (2023).
Akaike, H. A new look at the statistical model identification. IEEE Trans. Autom. Control 19, 716–723 (2003).
Lepeu, G. et al. The critical dynamics of hippocampal seizures. Nat. Commun. 15, 6945 (2024).
Li, Z. et al. Synchronization stability of epileptic brain network with higher-order interactions. Chaos 35, 013137 (2025).
Baumann, F., Lorenz-Spreen, P., Sokolov, I. M. & Starnini, M. Modeling echo chambers and polarization dynamics in social networks. Phys. Rev. Lett. 124, 048301 (2020).
Ott, E. & Antonsen, T. M. Low-dimensional behavior of large systems of globally coupled oscillators. Chaos 18, 037113 (2008).
Menck, P. J., Heitzig, J., Marwan, N. & Kurths, J. How basin stability complements the linear-stability paradigm. Nat. Phys. 9, 89–92 (2013).
Ma, J., Liu, Q., Xu, Y. & Kurths, J. Early warning of noise-induced catastrophic high-amplitude oscillations in an airfoil model. Chaos 32, 033119 (2022).
Zan, W., Jia, W. & Xu, Y. Reliability of dynamical systems with combined Gaussian and Poisson white noise via path integral method. Probab. Eng. Mech. 68, 103252 (2022).
Wang, Z., Yang, W., Zhang, X. & Xu, Y. Stochastic-resonance chimeras in coupled FHN neurons with α-stable lévy noise. Phys. A 682, 131146 (2025).
Acknowledgements
This work is supported by the Key International (Regional) Joint Research Program of the National Natural Science Foundation (NSF) of China under Grant No. 12120101002. D. Zhao thanks the Sino-German (CSC-DAAD) Postdoc Scholarship Program.
Author information
Authors and Affiliations
Contributions
D.Z. and Y.X. conceived and designed the research. D.Z. performed the main calculations and data analysis. J.K. and N.M. contributed to the development of the methodology and the interpretation of the results. D.Z. wrote the original draft of the manuscript. J.K., N.M., and Y.X. supervised the work and critically revised the manuscript. All authors discussed the results and approved the final version of the manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Peer review
Peer review information
Communications Physics thanks Sarika Jalan, Zhigang Zheng and the other anonymous reviewer(s) for their contribution to the peer review of this work.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.
About this article
Cite this article
Zhao, D., Kurths, J., Marwan, N. et al. Synchronization transitions and spike dynamics in a higher-order Kuramoto model with Lévy noise. Commun Phys (2026). https://doi.org/10.1038/s42005-026-02560-4
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s42005-026-02560-4
