Coupled nonlinear oscillators below the synchronization threshold: Relaxation by generalized Landau damping

We analyze a model of globally coupled nonlinear oscillators with randomly distributed frequencies. Twenty-five years ago it was conjectured that, for coupling strengths below a certain threshold, this system would always relax to an incoherent state. We prove this conjecture for the system linearized about the incoherent state, for frequency distributions with compact support. The relaxation is exponentially fast at intermediate times but slower than exponential at long times. The decay mechanism is remarkably similar to Landau damping in plasmas, even though the model was originally inspired by biological rhythms.