  We describe how the transition to synchronization in a system of globally
coupled Stuart-Landau oscillators changes from continuous to discontinuous when
the nature of the coupling is moved from diffusive to reactive. We explain this
drastic qualitative change as resulting from the co-existence of a particular
synchronized macrostate together with the trivial incoherent macrostate, in a
range of parameter values for which the later is linearly stable. In contrast
to the paradigmatic Kuramoto model, this particular state observed at the
synchronization transition contains a finite, non-vanishing number of
synchronized oscillators, which results in a discontinuous transition. We
consider successively two situations where either a fully synchronized state or
a partially synchronized state exist at the transition. Thermodynamic limit and
finite size effects are briefly discussed, as well as connections with recently
observed discontinuous transitions.
