Synchronization Dynamics in the Presence of Coupling Delays and Phase Shifts

In systems of coupled oscillators, the effects of complex signaling can be captured by time delays and phase shifts. Here, we show how time delays and phase shifts lead to different oscillator dynamics and how synchronization rates can be regulated by substituting time delays by phase shifts at constant collective frequency. For spatially extended systems with time delays, we show that fastest synchronization can occur for intermediate wavelengths, giving rise to novel synchronization scenarios.