A direct complex-domain stability criterion for synchronization of complex-valued dynamical networks

We present a direct criterion in the complex domain for analyzing the local exponential synchronization of complex networks of complex-valued systems. A standard procedure for analyzing synchronization involves splitting or transforming the complex-valued states into real and imaginary parts in the real domain, but this doubles the dimension of the system. However, this transformation obscures the richness of the complex system behavior. To address this, we propose a direct criterion in the complex domain Cn for network synchronization. This criterion preserves the system dimension, as well as its geometric and algebraic structure. Moreover, the criterion applies to both holomorphic and non-holomorphic systems, which means that it considers the effects of the complex conjugate states. The stability criterion can be viewed as a balance among three components: the local dissipation of the node dynamics, the destabilizing influence of the non-holomorphic part, which could induce instability, and the stabilizing effect of the diffusive coupling. We present two numerical examples to corroborate the criterion for exponential synchronization. One considers the synchronization of a network of Hamiltonian holomorphic oscillators, and the other presents the synchronization of a network of non-holomorphic Lorenz chaotic systems. The local direct synchronization criterion here presented is an alternative approach to studying synchronization of complex-valued systems, where the preservation of complex dynamic features is crucial.