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We study the synchronization behavior of Stuart-Landau oscillators coupled with delay, using analytical and numerical methods. We compare the dynamics of one oscillator with delayed feedback, two mutually oscillators coupled with delay, and two delay-coupled elements with feedback. Taking only the phase dynamics into account, no chaotic dynamics has been observed. Moreover, the stability of the symmetric (identical synchronization) solution is the same in each of the three studied networks of delay-coupled elements. When allowing variable oscillation amplitude, the delay can induce amplitude instabilities. We provide analytical proof that, in case of two mutually coupled elements, the onset of an amplitude instability is accompanied by a symmetry breaking, leading to the in lasers observed leader-laggard behavior in the chaotic regime. Adding self-feedback (with the same strength and delay as the coupling), stabilizes the system in transverse directionO.D. acknowledges the Research Foundation Flanders (FWO-Vlaanderen) for a fellowship and for project support.\udR.V. acknowledges the support of the Hertie Foundation. This work was partially supported by the\udInteruniversity Attraction Poles program of the Belgian Science Policy Office, under grant IAP VI-10 ”photonics@\udbe” and by the European Community Project GABA (FP6-NEST contract 043309).Peer reviewe

Synchronization and symmetry breaking of delay-coupled oscillators: on the role of phase and amplitude instabilities