Singular-Perturbations-Based Analysis of Dynamic Consensus in Directed Networks of Heterogeneous Nonlinear Systems

We investigate conditions under which heterogeneous nonlinear systems, interconnected over a directed static network, may achieve synchrony. Due to the network's heterogeneity, complete synchronization is impossible in general, but an emergent dynamics arises. This may be characterized by two dynamical systems evolving in two timescales. The first, which is “slow,” corresponds to the dynamics of the network on the synchronization manifold. The second, which is “fast,” corresponds to that of the synchronization errors. We present a framework to analyze the emergent dynamics based on the behavior of the slow dynamics. First, we give conditions under which if the slow dynamics admits a globally asymptotically stable (GAS) equilibrium, so do the networked systems. Second, we give conditions under which if the slow dynamics admits an asymptotically stable orbit and a single unstable equilibrium point, there exists a unique periodic orbit that is almost-GAS. The emergent behavior is thus clear; the systems asymptotically synchronize in frequency, and, in the limit, as the coupling strength grows unboundedly, the emergent dynamics approaches that of the slow system. Our analysis is established using singular-perturbations theory. In that regard, we also contribute with original statements on the stability of disconnected invariant sets and limit cycles for systems in singular-perturbation form.