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 <inline-formula> <tex-math notation="LaTeX">$\varepsilon $ </tex-math></inline-formula>-dependent Lyapunov functions. For a given upper bound of the singular perturbation parameter <inline-formula> <tex-math notation="LaTeX">$\varepsilon $ </tex-math></inline-formula>, it is proved that the exponential decay rate of the synchronization error can be guaranteed to be independent of the value of <inline-formula> <tex-math notation="LaTeX">$\varepsilon $ </tex-math></inline-formula>. In this way, criteria for local and global exponential synchronization are established. The allowable upper bound of <inline-formula> <tex-math notation="LaTeX">$\varepsilon $ </tex-math></inline-formula> such that the synchronizability of the considered two-time-scale network is retained can be obtained by solving a set of <inline-formula> <tex-math notation="LaTeX">$\varepsilon $ </tex-math></inline-formula>-dependent matrix inequalities. Finally, the efficiency of the proposed time-scale-dependent coupling strategy is demonstrated through numerical simulations.