Three Time Scales Modeling of the Undirected Clustered Network

Dynamics evolving on different time scales appear naturally in networks of clusters. In this paper, we emphasize that the synchronization of interconnected agents partitioned in clusters involves three time-scales. Using a coordinate transformation, we reformulate the overall closed-loop clustered network dynamics in new coordinates describing the mean-field dynamics, the intra-cluster and the inter-cluster synchronization error dynamics. Then, under realistic assumptions, we show that the network dynamics can be represented in a two-parameter standard singular perturbation form in the new coordinate. The mean-field dynamics, which is the network’s long-term behavior, evolve on the slowest time-scale. The intra-cluster error dynamics, which characterizes the synchronization inside clusters, evolve on the fastest time scale. Finally, the inter-cluster error dynamics, which characterizes the synchronization between clusters, is fast with respect to the mean-field one and slow with respect to the intra-cluster one. Using the two-parameter singular perturbation form, we establish approximations of the variables using standard techniques based on Tychonoff’s theorem. Numerical simulations validate our theoretical results.