Exotic Symmetry in Network Dynamical Systems

Network dynamical systems are highly prominent as models throughout science and engineering, yet at the same time they form a rich class of examples of theoretically important dynamical behavior. However, beyond statistical and numerical results, not a lot of techniques exist for elucidating the role of connectivity structure in shaping this behavior. The main reason for this is that most established techniques from dynamical systems theory involve coordinate changes that inevitable obscure the network structure. There are some notable exceptions to this though; networks with symmetry can be analyzed using these techniques, precisely because we know how to deal with the symmetry.  But what about non-symmetric networks? We present a way of translating a whole range of geometric network features, as well as network structure itself in a large class of examples, using the algebraic notion of quiver symmetry. We then show that quiver symmetry can be preserved in a wide range of dynamical techniques, most notably reduction techniques, thus leading to new methods that are tailor-made to the network setting. The main focus will be on centre manifold reduction, synchrony-breaking bifurcations and phase reduction.