From Delay to Inertia and Triadic Interactions: A Predictive Model for Time-Delayed Oscillator Networks

TL;DR

This paper introduces a second-order predictive reduction for time-delayed oscillator networks, capturing complex dynamics like chimeras and splay states through a delay-free model with inertia and triadic interactions.

Contribution

It develops a universal, parameter-explicit reduced model that accurately predicts high-dimensional delay-induced phenomena in oscillator networks, extending to amplitude-phase systems.

Findings

Accurately predicts nontrivial attractors and collective states.

Reveals roles of inertia and triadic interactions in pattern formation.

Applicable to arbitrary network topologies and oscillator types.

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 Letter, 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.