Synchronization and Amplitude Death in Coupled Limit Cycle Oscillators with Time Delays

A pair of coupled van der Pol oscillators is considered whose interaction involves time delays. Using averaging theory, synchronous periodic solutions are determined, where the phase difference between the oscillators remains fixed in time. Parameter ranges for the coupling strength and delay are calculated such that the oscillators exhibit stable in-phase or anti-phase synchronized oscillations with identical amplitudes. It is shown that for any value of the delay there exists either an in-phase or an anti-phase synchronized solution which is asymptotically stable. In addition, parameter values are found for which the system is multistable, where both type of oscillations can be observed depending on the initial conditions, or it experiences amplitude death, where all oscillations are quenched.