On Global Asymptotic Stability of Heterogeneous Modular Networks with Three Time-Scales

We analyze the collective behavior of relatively large networks with the characteristic that nodes may be compartmentalized in modules. These are groups of systems that may be regarded as subnetworks in which each group of nodes achieves consensus with a certain rapidity. The dynamics of such networks exhibit, at least, two time-scales, for which singular perturbation methods may be used to assess the overall behavior. In this paper, we demonstrate that there are actually three natural time-scales and, accordingly, three interconnected dynamical systems with distinct speeds of convergence coexist. Our main statement establishes conditions for global asymptotic stability of the origin in such networked systems. In particular, we focus on bilinear heterogeneous systems and provide an illustrative example concerning chaotic oscillators.