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  • Open Access

Combined framework for synchronization with distributed delays and noise in Kuramoto-Daido oscillators

Samir Sahoo* and Antonio Palacios†

  • *Contact author: samirkumsahoo@gmail.com
  • †Contact author: apalacios@sdsu.edu
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Phys. Rev. Research 8, 043038 – Published 13 October, 2026

DOI: https://doi.org/10.1103/v5cl-krph

Abstract

We derive the characteristic equation governing the linear stability of the incoherent state in delayed noisy Kuramoto-Daido oscillator ensembles with general periodic coupling. The calculation is formulated for arbitrary normalized delay kernels and natural-frequency distributions. For a Lorentzian frequency distribution and first-harmonic coupling, the characteristic equation reduces to a closed dispersion relation in which phase diffusion and frequency heterogeneity enter through the damping combination D+Δ, while the delay kernel enters through its Laplace transform. The neutral conditions recover the established fixed- and distributed-delay limits and are evaluated explicitly for Gamma-distributed delays. We then derive the reduced phase dynamics of an overdamped Josephson-junction array coupled through a delayed common load. A finite-dc-load admittance produces a uniform drift that corresponds to the constant Fourier component of the reduced coupling function. When the ac coupling and dc-induced drift vary with the same load parameter, the resulting effective-frequency shift displaces the neutral stability boundary. Finite-N simulations of the reduced phase equations agree with the continuum prediction over the parameter range considered.

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References (28)

  1. A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences (Cambridge University Press, Cambridge, 2001).
  2. J. A. Acebrón, L. L. Bonilla, C. J. Pérez Vicente, F. Ritort, and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys. 77, 137 (2005).
  3. F. A. Rodrigues, T. K. DM Peron, P. Ji, and J. Kurths, The Kuramoto model in complex networks, Phys. Rep. 610, 1 (2016).
  4. H. Sompolinsky, D. Golomb, and D. Kleinfeld, Global processing of visual stimuli in a neural network of coupled oscillators, Proc. Natl. Acad. Sci. USA 87, 7200 (1990).
  5. E. M. Izhikevich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting (MIT Press, Cambridge, MA, 2007).
  6. S. Yanchuk and M. Wolfrum, A multiple time scale approach to the stability of external cavity modes in the Lang–Kobayashi system using the limit of large delay, SIAM J. Appl. Dyn. Syst. 9, 519 (2010).
  7. K. Wiesenfeld, P. Colet, and S. H. Strogatz, Synchronization transitions in a disordered Josephson series array, Phys. Rev. Lett. 76, 404 (1996).
  8. A. E. Motter, S. A. Myers, M. Anghel, and T. Nishikawa, Spontaneous synchrony in power-grid networks, Nat. Phys. 9, 191 (2013).
  9. Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence (Springer, Berlin, 1984).
  10. M. G. Earl and S. H. Strogatz, Synchronization in oscillator networks with delayed coupling: A stability criterion, Phys. Rev. E 67, 036204 (2003).
  11. S. Yanchuk and G. Giacomelli, Spatio-temporal phenomena in complex systems with time delays, J. Phys. A: Math. Theor. 50, 103001 (2017).
  12. H. Sakaguchi, Cooperative phenomena in coupled oscillator systems under external fields, Prog. Theor. Phys. 79, 39 (1988).
  13. S. H. Strogatz and R. E. Mirollo, Stability of incoherence in a population of coupled oscillators, J. Stat. Phys. 63, 613 (1991).
  14. L. L. Bonilla, J. C. Neu, and R. Spigler, Nonlinear stability of incoherence and collective synchronization in a population of coupled oscillators, J. Stat. Phys. 67, 313 (1992).
  15. J. D. Crawford, Amplitude expansions for instabilities in populations of globally-coupled oscillators, J. Stat. Phys. 74, 1047 (1994).
  16. H. Sakaguchi and Y. Kuramoto, A soluble active rotator model showing phase transitions via mutual entrainment, Prog. Theor. Phys. 76, 576 (1986).
  17. H. Daido, Order function and macroscopic mutual entrainment in uniformly coupled limit-cycle oscillators, Prog. Theor. Phys. 88, 1213 (1992).
  18. H. Daido, Onset of cooperative entrainment in limit-cycle oscillators with uniform all-to-all interactions: Bifurcation of the order function, Physica D 91, 24 (1996).
  19. J. D. Crawford and K. T. R. Davies, Synchronization of globally coupled phase oscillators: Singularities and scaling for general couplings, Physica D 125, 1 (1999).
  20. H. Chiba, Linear stability of the incoherent solution and the transition formula for the Kuramoto–Daido model, RIMS Kôkyûroku Bessatsu B 21, 109 (2010).
  21. M. K. S. Yeung and S. H. Strogatz, Time delay in the Kuramoto model of coupled oscillators, Phys. Rev. Lett. 82, 648 (1999).
  22. W. S. Lee, E. Ott, and T. M. Antonsen, Large coupled oscillator systems with heterogeneous interaction delays, Phys. Rev. Lett. 103, 044101 (2009).
  23. K. Wiesenfeld and J. W. Swift, Averaged equations for Josephson junction series arrays, Phys. Rev. E 51, 1020 (1995).
  24. B. C. Daniels, S. T. M. Dissanayake, and B. R. Trees, Synchronization of coupled rotators: Josephson junction ladders and the locally coupled Kuramoto model, Phys. Rev. E 67, 026216 (2003).
  25. J. Turtle, P.-L. Buono, A. Palacios, C. Dabrowski, V. In, and P. Longhini, Synchronization of spin torque nano-oscillators, Phys. Rev. B 95, 144412 (2017).
  26. B. Sturgis-Jensen, P.-L. Buono, A. Palacios, J. Turtle, V. In, and P. Longhini, On the synchronization phenomenon of a parallel array of spin torque nano-oscillators, Physica D 396, 70 (2019).
  27. T. Levasseur, A. Palacios, S. Sharan, and V. In, Beam steering and signal amplification through feedforward networks. Part I: Transmission, Int. J. Bifurc. Chaos 32, 2230034 (2022).
  28. T. Levasseur, A. Palacios, S. Sharan, and V. In, Beam steering and signal amplification through feedforward networks: Part II reception, Int. J. Bifurc. Chaos 35, 2530004 (2025).

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Phys. Rev. Research 8, 043038– Published 13 October, 2026
  • Received 29 May 2026
  • Accepted 18 September 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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