Forced symmetry-breaking in networks with dihedral symmetry.

Adaptation in complex systems implies a natural ability to change. In networks, adaptation may include a change in structural connectivity, which can lead to a change in collective behavior. When dihedral symmetry is present, i.e., rotations and reflections of a regular polygon, it is well-known that traveling and standing waves occur, generically, via spontaneous symmetry-breaking Hopf bifurcations. While synchronization appears via standard, symmetry-preserving, Hopf bifurcations. In these cases, the symmetries of the network equations do not change even though the bifurcating solutions may lose symmetry as parameters are varied. But when they do, possibly due to adaptation, there is, however, little knowledge of what happens to those patterns. Here, we choose to investigate the effects of forced-breaking the rotation symmetry of a network with (unperturbed) dihedral symmetry. We study, in particular, the changes in the region of existence and stability of the unperturbed patterns-traveling and standing waves and synchronization.