Synchronization analysis of two-time-scale nonlinear complex networks with time-scale-dependent coupling

In this paper, a time-scale-dependent coupling scheme for two-time-scale nonlinear complex networks is proposed. According to this scheme, the inner coupling matrices are related to the fast dynamics of individual subsystems, but are no longer time-scale-independent. Designing time-scale-dependent inner coupling matrices is motivated by the fact that the difference of time scales is an essential feature of modular architecture of two-time-scale systems. Under the novel coupling framework, the previous assumption on individual two-time-scale subsystems that the fast dynamics must be exponentially stable can be removed. The idea of time-scale separation is employed to analyze the stability of synchronization error systems via weighted ϵ-dependent Lyapunov functions. For a given upper bound of the singular perturbation parameter ϵ, it is proved that the exponential decay rate of the synchronization error can be guaranteed to be independent of the value of ϵ. In this way, criteria for local and global exponential synchronization are established. The allowable upper bound of ϵ such that the synchronizability of the considered two-time-scale network is retained can be obtained by solving a set of ϵ-dependent matrix inequalities. Finally, the efficiency of the proposed time-scale-dependent coupling strategy is demonstrated through numerical simulations.