We generalize the Kuramoto model of coupled oscillators to allow time-delayed interactions. New phenomena include bistability between synchronized and incoherent states, and unsteady solutions with time-dependent order parameters. We derive exact formulas for the stability boundaries of the incoherent and synchronized states, as a function of the delay, in the special case where the oscillators are identical. The experimental implications of the model are discussed for populations of chirping crickets, where the finite speed of sound causes communication delays, and for physical systems such as coupled phase-locked loops or lasers.
M. K. Stephen Yeung and Steven H. Strogatz. Time Delay in the Kuramoto Model of Coupled Oscillators. Phys. Rev. Lett. 82, 648 (1999). Copyright American Physical Society.