Synchronization of nearly-identical dynamical systems. I Size instability

Abstract: We study the generalized synchronization and its stability using master stability function (MSF), in a network of coupled nearly identical dynamical systems. We extend the MSF approach for the case of degenerate eigenvalues of the coupling matrix. Using the MSF we study the size-instability in star and ring networks for coupled nearly identical dynamical systems. In the star network of coupled R\"ossler systems we show that the critical size beyond which synchronization is unstable, can be increased by having a larger frequency for the central node of the star. For the ring network we show that the critical size is not significantly affected by parameter variations. The results are verified by explicit numerical calculations.