2006/03/01 by Fernando Antoneli, FERNANDO ANTONELI, Ian Stewart +1 · 2 citations
Computer Science · Biochemistry, Genetics and Molecular Biology · Physics and Astronomy · #Nonlinear Dynamics and Pattern Formation #Microtubule and mitosis dynamics #Advanced Differential Geometry Research
paper · doi:10.1142/s0218127406015167
Equivariant dynamical systems possess canonical flow-invariant subspaces, the fixed-point spaces of subgroups of the symmetry group. These subspaces classify possible types of symmetry-breaking. Coupled cell networks, determined by a symmetry groupoid, also possess canonical flow-invariant subspaces, the balanced polydiagonals. These subspaces classify possible types of synchrony-breaking, and correspond to balanced colorings of the cells. A class of dynamical systems that is common to both theories comprises networks that are symmetric under the action of a group Γ of permutations of the nodes ("cells"). We investigate connections between balanced polydiagonals and fixed-point spaces for such networks, showing that in general they can be different. In particular, we consider rings of ten and twelve cells with both nearest and next-nearest neighbor coupling, showing that exotic balanced polydiagonals — ones that are not fixed-point spaces — can occur for such networks. We also prove the "folk theorem" that in any Γ-equivariant dynamical system on R k the only flow-invariant subspaces are the fixed-point spaces of subgroups of Γ.