Synchronization in coupled time-delayed systems

Synchronization in a bidirectionally coupled system with a finite time-delay in coupling is studied using Josephson junction as a model system. Using the LyapunovKrasovskii method a sufficient condition for the stability of synchronization is obtained and it is found to be independent of the time-delay. The sum of the Lyapunov exponents transverse to the synchronization manifold is evaluated. Numerical simulations show that the system attains periodic nature for low values of time-delay and becomes chaotic as time-delay is increased while synchronization remains unaffected in the parameter range considered. A phase difference among the applied fields along with a time-delay takes the system from chaotic motion to periodic motion while the effect of frequency detuning on the system remain unchanged by time-delay.