Synchronization in an array of linearly coupled dynamical systems

This paper gives sufficient conditions for an array of linearly coupled systems to synchronize. A typical result states that the array will synchronize if the nonzero eigenvalues of the coupling matrix have real parts that are negative enough. In particular, we show that the intuitive idea that strong enough mutual diffusive coupling will synchronize an array of identical cells is true in general. Sufficient conditions for synchronization for several coupling configurations will be considered. For coupling that leaves the array decoupled at the synchronized state, the cells each follow their natural uncoupled dynamics at the synchronized state. We illustrate this with an array of chaotic oscillators. Extensions of these results to general coupling are discussed.>