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.