4. Synchronisation and Emergent Behaviour in Networks of Heterogeneous Systems: A Control Theory Perspective

Generally speaking, for a network of interconnected systems, synchronisation consists in the mutual coordination of the systems’ motions to reach a common behaviour. For homogeneous systems that have identical dynamics this typically consists in asymptotically stabilising a common equilibrium set. In the case of heterogeneous networks, in which systems may have different parameters and even different dynamics, there may exist no common equilibrium but an emergent behaviour arises. Inherent to the network, this is determined by the connection graph but it is independent of the interconnection strength. Thus, the dynamic behaviour of the networked systems is fully characterised in terms of two properties whose study may be recast in the domain of stability theory through the analysis of two interconnected dynamical systems evolving in orthogonal spaces: the emergent dynamics and the synchronisation errors relative to the common behaviour. Based on this premise, we present some results on robust stability by which one may assess the conditions for practical asymptotic synchronisation of networked systems. As an illustration, we broach a brief case-study on mutual synchronisation of heterogeneous chaotic oscillators.