11. Synchronization and Consensus

Synchronization is one of the fundamental aspects of self-organization in networks of systems. More generally, the emergence of macroscopic states is frequently encountered in dynamical systems when one starts to study coupling effects. Well-known examples include synchronization of oscillators, the emergence of consensus states in models that describe the opinion dynamics of social networks or multiagent systems, or flocking phenomena in biological networks such as swarms of birds or a school of fish. In all these different network models the dynamics of the individual states may “cluster” together or “synchronize” toward a common state that exhibits the system with a unique characteristic identity. The analysis and control of such synchronized states thus becomes an interesting new task for the design of networks. The phenomenon of synchrony was apparently noticed first by Huygens, who was, alongside his scientific activity, also a clock maker. Huygens noticed that two pendulum clocks hanging on a wall tend to synchronize. With time, a multitude of synchronization phenomena were observed in different fields, including, for example, ciliary beating in biology, laser physics, and the firing of neurons as in Parkinson disease. Although most realistic models are of course nonlinear, it appears to be of fundamental interest to explore these issues first in the simplified context of linear systems theory.