M. Golubitsky and I. Stewart
Rigid patterns of synchrony for equilibria and periodic cycles in network dynamics
Chaos.
Submitted.
We survey general results relating patterns of synchrony
to network architecture, applying the formalism of
coupled cell systems. We also discuss patterns of phase-locking
for periodic states, where nodes have identical waveforms but
regularly spaced phases. We focus on rigid patterns,
which are not changed by small perturbations
of the differential equation. Symmetry is one mechanism that creates
patterns of synchrony and phase-locking. In general networks
there is another: balanced colorings of the cells. A
symmetric network may have anomalous patterns of synchrony and
phase-locking that are not consequences of symmetry.
We introduce basic notions on coupled cell networks and their
associated systems of admissible differential equations.
Periodic states also possess spatio-temporal symmetries,
leading to phase relations; these are classified by the H/K Theorem
and its analog for general networks.