Coupled Networks, Patterns and Complexity - Abstract
Pade, Jan Philipp
It is known that for a homogeneous, unidirectional ring of Stuart-Landau oscillators the trivial equilibrium undergoes a supercritical Hopf bifurcation, followed by a sequence of Hopf bifurcations. Periodic solutions that emerge from these bifurcations, although born unstable, gain stability when the bifurcation parameter is increased. In spatially extended diffusive systems this phenomenon is well-known under the name of Eckhaus scenario, but it was rarely observed in ODEs that do not extend to infinite dimensional systems in a natural manner. We show that the Eckhaus diagram persists under small perturbations of the network topology, i.e. insertion of a crosslink. For large perturbations of the network topology we employ a perturbation theoretic argument to show that the coexisting periodic solutions split up in two groups of unstable and stable modes.