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  • Open Access

Coherence properties of collective modes in ensembles of oscillators

Arkady Pikovsky1,*, Franco Bagnoli2,3,†, and Stefano Iubini4,3,‡

  • *Contact author: pikovsky@uni-potsdam.de
  • †Contact author: franco.bagnoli@unifi.it
  • ‡Contact author: stefano.iubini@cnr.it
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Phys. Rev. E 113, 054209 – Published 11 May, 2026

DOI: https://doi.org/10.1103/lrp8-4wxh

Abstract

Synchronization transitions in oscillatory networks often manifest as the emergence of a periodic global mode. While perfect in the thermodynamic limit, this mode fluctuates for finite ensembles. We characterize the coherence of this mode in terms of the phase diffusion constant. In several examples, we always observed normal diffusion, but the dependence of the diffusion constant on the system size D∼N−μ depends on the nature of coupled units. For coupled chaotic systems, we found μ=1, while for coupled periodic oscillators we observed, depending on the particular model, values between 2 and 2.7. These large values of the power index are attributed to the size dependence of collective chaos in the finite ensemble, which disappears in the thermodynamic limit. We also show that in the standard Kuramoto model for a symmetric set of frequencies, there is an additional transition to a symmetric chaotic state with vanishing diffusion of the global phase.

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Phys. Rev. E 113, 054209– Published 11 May, 2026
  • Received 9 January 2026
  • Accepted 22 April 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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