Higher-order phase reduction captures delay-dependent synchronization phenomena in physical oscillator networks

Coupled oscillators with time-delayed network interactions are critical to understand synchronization phenomena in many physical systems.

Phase reductions to finite-dimensional phase oscillator networks yield explicit insights into their dynamics.

However, first-order phase approximations - in which the time delay acts as a phase shift - fail to capture the delay-dependence of synchronization.

We develop a systematic approach to derive phase reductions for delay-coupled oscillators to higher order.

Beyond first order, already a second-order phase reduction captures delay-induced synchronization as demonstrated in coupled Stuart-Landau oscillators and experiments with delay-coupled electrochemical oscillators.

Our results establish a general mechanism by which time delays reshape synchronization phenomena and reveal intrinsic limitations of widely used reduced phase models, with implications for a broad range of oscillator networks.