M. Golubitsky, L. Matamba Messi and L.E. Spardy
Symmetry Types and Phase-Shift Synchrony in Networks
Physica D.
Submitted.
In this paper we discuss what is known about the classification of symmetry groups and patterns
of phase-shift synchrony for periodic solutions of coupled cell networks. Specifically, we compare
the lists of spatial and spatiotemporal symmetries of periodic solutions of admissible vector fields
to those of equivariant vector fields in the three cases of R^n (coupled equations), T^n
(coupled oscillators), and (R^k)^n where k ge 2 (coupled systems). To do this we use
the H/K Theorem of Buono and Golubitsky applied to coupled equations and coupled systems
and prove the $H/K$ theorem in the case of coupled oscillators. Josic and Torok prove that
the H/K lists for equivariant vector fields and admissible vector fields are the same for transitive
coupled systems. We show that the corresponding theorem is false for coupled equations. We also
prove that the pairs of subgroups H supset K for coupled equations are contained in the pairs for
coupled oscillators which are contained in the pairs for coupled systems. Finally, we prove that
patterns of rigid phase-shift synchrony for coupled equations are contained in those of coupled
oscillators and those of coupled systems.