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From delay to inertia and triadic interactions: A predictive model for time-delayed oscillator networks

Lev A. Smirnov1, Vyacheslav O. Munyayev1, Maxim I. Bolotov1, and Igor Belykh2,*

  • *Contact author: ibelykh@gsu.edu
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Phys. Rev. Research 8, 033189 – Published 17 August, 2026

DOI: https://doi.org/10.1103/f5n5-8mr3

Abstract

Time-delayed oscillator networks underlie diverse biological and physical systems, yet standard first-order phase reductions fail to capture their high-dimensional collective dynamics. In this work, we develop a universal second-order predictive reduction for time-delayed Kuramoto-Daido networks that maps delayed one-dimensional phase dynamics to a delay-free network of two-dimensional rotators. Delay induces effective inertia and triadic interactions, yielding accurate predictions of nontrivial attractors and their collective-state statistics, including splay, cyclops, and chimera states. The reduction reveals a division of roles: Inertia organizes higher-dimensional dynamics, whereas triadic terms are crucial for lower-dimensional patterns such as chimeras. Applicable to arbitrary topology, higher harmonics, and intrinsic-frequency heterogeneity, it provides a compact, parameter-explicit reduced model. The same framework also extends to time-delayed amplitude-phase oscillator networks, including swarmalators, yielding analogous reduced equations with emergent inertia and triadic higher-order couplings. This unified and readily deployable description enables systematic prediction and analysis of delay-controlled collective dynamics across oscillator networks.

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Phys. Rev. Research 8, 033189– Published 17 August, 2026
  • Received 12 December 2025
  • Accepted 17 July 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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