Center manifolds of coupled cell networks

Many systems in science and technology are networks: they consist of nodes with connections between them. Examples include electronic circuits, power grids, neuronal networks, and metabolic systems. Such networks are usually modeled by coupled nonlinear maps or differential equations, that is, as network dynamical systems. Network dynamical systems often behave very differently from regular dynamical systems that do not possess the structure of a network, and the interaction between the nodes of a network can spark surprising emergent behavior. An example is synchronization, the process by which neurons fire simultaneously and social consensus is reached. This paper is concerned with synchrony breaking, the phenomenon that less synchronous solutions emerge from more synchronous solutions as model parameters vary. It turns out that synchrony breaking often occurs via remarkable anomalous bifurcation scenarios. As an explanation for this it has been noted that homogeneous networks can be realized as quotient networks of so-called fundamental networks. The class of admissible dynamical systems for these fundamental networks is equal to the class of equivariant (symmetric) dynamical systems of the regular representation of a monoid (a monoid is an algebraic semigroup with unit). Using this geometric insight, we set up a framework for center manifold reduction in fundamental networks and their quotients. We then use this machinery to classify generic synchrony breaking bifurcations in three example networks with identical spectral properties and identical robust synchrony spaces.