Abstract

Network dynamical systems show intricate dynamical behavior that is highly unusual for general ODEs. Examples include invariant (synchrony) spaces, eigenvalue degeneracies, and elaborate synchrony-breaking bifurcations. These phenomena are reminiscent of those seen in systems with group symmetry. Indeed, it is well-understood how a group of (linear) transformations that send solutions to solutions impacts the dynamics, and the corresponding field of equivariant dynamics is well-established.

 

Unfortunately, there are many network structures that have no classical group symmetry at all, yet which still show such highly intricate dynamical behavior. To make matters worse, many established techniques from bifurcation theory cannot deal with network structure, as they involve coordinate changes that obscure the connectivity structure.

 

We present a solution by showing that many geometric features of a network — and often the precise network structure itself — can still be seen as a form of symmetry, provided one abandons the notion of a group. Instead, the symmetries at play represent more "exotic" algebraic structures, such as semigroups, monoids, quivers and categories. We then show that such symmetries may be preserved in various complexity reduction techniques, leading to dynamical techniques that are tailor-made for networks. This is based on joint work with Sören von der Gracht and Bob Rink. 

WebEx Link: https://nationaltaiwanuniversity-ksz.my.webex.com/nationaltaiwanuniversity-ksz.my/j.php?MTID=mb9c153d669efcc2d2e41f79052777f8f

Meeting number (access code): 2515 467 8749

Meeting password: g2Y8y323ENc

 

Organizers: Chueh-Hsin Chang (CCU), Jia-Yuan Dai (NTHU), Bo-Chih Huang (CCU), Chih-Chiang Huang (CCU), Alejandro López-Nieto (NTU), Chang-Hong Wu (NYCU)