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