Effects of time delay on the synchronized states of globally coupled network

The effects of the time delay on the stability of different synchronized states of a globally coupled network are investigated. Conditions for the stability of the synchronized fixed points, synchronized periodic orbits, or synchronized chaos in a network of globally coupled chaotic smooth maps over a ring lattice with a homogeneous delay are derived analytically. Our analysis reveals that the stability properties of the synchronized dynamics are significantly different for odd and even time delays. The conditions for the stability of a synchronized fixed point and synchronized period-2 orbits for both odd and even delays are determined analytically. The range of parameter values for the stability of synchronized chaos has been calculated for a unit delay. All theoretical results are illustrated with the help of numerical examples.