Center manifolds of coupled cell networks

Dynamical systems with a network structure can display anomalous bifurcations as a generic phenomenon. 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 vector fields for these fundamental networks is equal to the class of equivariant vector fields of the regular representation of a monoid. Using this insight, we set up a framework for center manifold reduction in fundamental networks and their quotients. We then use this machinery to explain the difference in generic bifurcations between three example networks with identical spectral properties and identical robust synchrony spaces

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

Quiver representations and dimension reduction in dynamical systems

Dynamical systems often admit geometric properties that must be taken into account when studying their behaviour. We show that many such properties can be encoded by means of quiver representations. These properties include classical symmetry, hidden symmetry and feedforward structure, as well as subnetwork and quotient relations in network dynamical systems. A quiver equivariant dynamical system consists of a collection of dynamical systems with maps between them that send solutions to solutions. We prove that such quiver structures are preserved under Lyapunov-Schmidt reduction, center manifold reduction, and normal form reduction.Comment: Revised version, accepted in the SIAM Journal on Applied Dynamical Systems; 43 pages, 10 figure

Quiver representations and dimension reduction in dynamical systems

Dynamical systems often admit geometric properties that must be taken into account when studying their behavior. We show that many such properties can be encoded by means of quiver representations. These properties include classical symmetry, hidden symmetry, and feedforward structure, as well as subnetwork and quotient relations in network dynamical systems. A quiver equivariant dynamical system consists of a collection of dynamical systems with maps between them that send solutions to solutions. We prove that such quiver structures are preserved under Lyapunov-Schmidt reduction, center manifold reduction, and normal form reduction