Symmetry types and phase-shift synchrony in networks

Abstract

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 ≥ 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. Josić and Török [17] 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 ⊃ 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.

Overdetermined ODEs and Rigid Periodic States in Network Dynamics

“…13 Cyclic Automorphisms and the H/K Theorem It is known that if conjectures (a, b, c, d) are valid for a network G, which we have proved is the case for strongly hyperbolic periodic orbits, then there are important consequences for the combinatorial structure of G. In Golubitsky et al [34] and [72] it is proved that, on the assumption that these conjectures are valid for a given network G, there is a natural network analogue of the H/K Theorem of Buono and Golubitsky [14]; see also Golubitsky and Stewart [36] and Golubitsky et al [31].…”

“…However, equivariant maps need not be admissible [5,Section 3.1]. Examples of synchrony and phase patterns of these kinds can be found in many papers, for instance [4,5,6], Buono and Golubitsky [14], Golubitsky et al [31,32,37,56,69].…”

Overdetermined ODEs and rigid periodic states in network dynamics

“…The methods employed in this paper probably generalise to manifolds. However, Golubitsky et al [31] show that the topology of node spaces can change the list of possible phase patterns in the H=K Theorem, so it should not be assumed that all of the results proved here automatically remain valid when node spaces are manifolds, or that they are independent of their topology.…”

“…However, equivariant maps need not be admissible [5,Section 3.1]. Examples of synchrony and phase patterns of these kinds can be found in many papers, for instance [4][5][6], Buono and Golubitsky [14], Golubitsky et al [31,32,37,57,70].…”

Detecting coexisting oscillatory patterns in delay coupled Lur’e systems

“…It is worth mentioning that the relative phases in ring networks can be extracted from the symmetry argument. 39,40 An advantage of the MHB approach is that it is not limited only to the networks with a ring structure and aiming at computing the whole oscillatory profile (i.e., the offset, amplitudes, phases, and frequency) in one shot. Another large advantage of the multivariable harmonic balance method is the small number of parameters of the harmonic approximation looked for (at the price of a possible over-approximation of the solution's profile).…”