We study the destabilization mechanism in a unidirectional ring of identical oscillators,
perturbed by the introduction of a long-range connection. It is known that for a
homogeneous, unidirectional ring of identical Stuart-Landau oscillators the trivial
equilibrium undergoes a sequence of Hopf bifurcations eventually leading to the
coexistence of multiple stable periodic states resembling the Eckhaus scenario. We show
that this destabilization scenario persists under small non-local perturbations. In this
case, the Eckhaus line is modulated according to certain resonance conditions. In the case
when the shortcut is strong, we show that the coexisting periodic solutions split up into
two groups. The first group consists of orbits which are unstable for all parameter
values, while the other one shows the classical Eckhaus behavior.