Symmetry types and phase-shift synchrony in networks

• Comparisons between H/KH/K lists for equations, oscillators, and systems are made.
• Comparisons of phase-shift patterns are made.
• The H/KH/K theorem for periodic solutions of coupled oscillators is valid.
• The H/KH/K lists for equivariant and admissible maps are not equal for coupled equations.

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 RnRn (coupled equations), TnTn (coupled oscillators), and (Rk)n(Rk)n where k≥2k≥2 (coupled systems). To do this we use the H/KH/K Theorem of Buono and Golubitsky (2001) applied to coupled equations and coupled systems and prove the H/KH/K theorem in the case of coupled oscillators. Josić and Török (2006) prove that the H/KH/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⊃KH⊃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.