Higher-order phase reduction for delay-coupled oscillators beyond the phase-shift approximation

Network interactions between dynamical units are often subject to time delay. We develop a phase reduction method for delay-coupled oscillator networks. The method is based on rewriting the delay-differential equation as an ordinary differential equation coupled with a transport equation, expanding in the coupling strength, and solving the resulting equations order-by-order. This approach yields an approximation of the finite-dimensional phase dynamics to arbitrary order. While in the first-order approximation the time delay acts as a phase shift as expected, the higher-order phase reduction generally displays a less trivial dependence on the delay. In particular, exploiting second-order phase reduction, we prove the existence of a region of bistability in the synchronization dynamics of two delay-coupled Stuart-Landau oscillators.