From Delay to Inertia and Triadic Interactions: A Reduction of Coupled Time-Delayed Oscillators

Abstract: Time-delayed phase-oscillator networks model diverse biological and physical systems, yet standard first-order phase reductions cannot adequately capture their high-dimensional collective dynamics. In this Letter, we develop a second-order reduction for a broad class of time-delayed Kuramoto-Daido networks, transforming the original delayed system of one-dimensional phase oscillators into a delay-free network of two-dimensional rotators. The resulting model shows that coupling delay generates inertial terms in the intrinsic dynamics and higher-order (triadic) interactions, and it accurately predicts the emergence of complex collective patterns such as splay, cyclops, and chimera states. The reduction further reveals a qualitative division of roles: time delay acts primarily as effective inertia for higher-dimensional dynamics, including splay states, whereas the induced triadic interactions are decisive for lower-dimensional patterns such as chimeras. The method applies to networks with arbitrary topology, higher-harmonic coupling, and intrinsic-frequency heterogeneity, yielding a compact, parameter-explicit reduced model. This universal reduced description of time-delayed oscillator networks opens the door to systematic prediction and analysis of nontrivial collective dynamics in delay-coupled systems.