Synchronization Phenomena

Complex systems are based on interacting local computational units may show non-trivial emerging behaviors. Examples are the time evolution of an infectious disease in a certain city that is mutually influenced by an ongoing outbreak of the same disease in another city, or the case of a neuron firing spontaneously while processing the effects of afferent axon potentials. A fundamental question is whether the time evolutions of interacting local units remain dynamically independent of each other, or whether they will change their states simultaneously, following identical rhythms. This is the notion of synchronization, which we will study throughout this chapter. Starting with the paradigmatic Kuramoto model we will learn that synchronization processes may be driven either by averaging dynamical variables, or through causal mutual influences. On the way, we will visit piecewise linear dynamical systems and the reference model for infectious diseases, the SIRS model.