Equivariant dynamical systems have canonical flow-invariant subspaces, the fixed point subspaces of subgroups, induced by symmetry; coupled cell systems have canonical flow-invariant subspaces, the polydiagonal subspaces of synchrony patterns, induced by coupling structure. In this talk, we review some similarities and differences between symmetry and synchrony in structured dynamical systems; give definitions of various forms of symmetry in coupled cell networks, and discuss to what extent one may extend the theory of equivariant dynamical systems to coupled cell systems.
The talk is based on the following joint scientific work with M. Aguiar:
• H. Ruan, A degree theory for coupled cell systems with quotient symmetries, Nonlinearity, accepted.
• M. Aguiar and H. Ruan, Interior symmetries and real multiple eigenvalues for homogeneous networks, SIAM J. Appl. Dynam. Sys, accepted.
• M. Aguiar and H. Ruan, Evolution of synchrony under combination of coupled cell networks, Nonlinearity, submitted.