  We present a theoretical study of synchronization dynamics in incoherently
pumped exciton-polariton condensates in coupled polariton traps. Our analysis
is based on a coupled-mode theory for the generalized Gross-Pitaevskii
equation, which employs an expansion in non-Hermitian, pump-dependent modes
appropriate for the pumped geometry. We find that polariton-polariton and
reservoir-polariton interactions play competing roles and lead to qualitatively
different synchronized phases of the coupled polariton modes as pumping power
is increased. Crucially, these interactions can also act against each other to
hinder synchronization. We map out a phase diagram and discuss the general
characteristics of these phases using a generalized Adler equation.
