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We study the destabilization mechanism in a unidirectional ring of identicaloscillators, perturbed by the introduction of a long-range connection. It isknown that for a homogeneous, unidirectional ring of identical Stuart-Landauoscillators the trivial equilibrium undergoes a sequence of Hopf bifurcationseventually leading to the coexistence of multiple stable periodic statesresembling the Eckhaus scenario. We show that this destabilization scenariopersists under small non-local perturbations. In this case, the Eckhaus line ismodulated according to certain resonance conditions. In the case when theshortcut is strong, we show that the coexisting periodic solutions split upinto two groups. The first group consists of orbits which are unstable for allparameter values, while the other one shows the classical Eckhaus behavior.

The dynamical impact of a shortcut in unidirectionally coupled rings of oscillators