Singular-perturbations-based analysis of synchronization in heterogeneous networks: A case-study

Abstract

In recent work, we laid the basis of an analysis framework for the study of heterogeneous networks. In essence, it is postulated that in a heterogeneous network a collective non-trivial behaviour arises, which may be modelled as a dynamical system itself. Then, we say that the networked systems synchronize or, more precisely, achieve dynamic consensus if they adopt this emergent behaviour. In this paper we consider the case-study of coupled Andronov-Hopf oscillators. We establish that the emergent dynamics, which is of the same nature as a single oscillator, is orbitally stable. Then, we show that the trajectories of the individual oscillators tend to a neighbourhood of the stable orbit. For the first time in the study of synchronization, the analysis is based on singular-perturbations theory; we show that the emergent dynamics corresponds to a slow system while the synchronization errors form a fast dynamics.