  We provide an analytical and numerical description of relaxation oscillations
in the nonresonantly pumped polariton condensate. The presented considerations
are based on the open dissipative Gross-Pitaevskii equation coupled to a pair
of rate equations. The evolution of the condensate density can be explained
qualitatively by studying the topology of the trajectory in phase space. We use
a fixed points analysis for the classification of the different regimes of
condensate dynamics, including fast stabilization, slow oscillations and
ultrashort pulse emission. We obtain an analytical condition for the occurrence
of relaxation oscillations. Continuous and pulsed condensate excitation
considered and we demonstrate that in the latter case the existence of the
second reservoir is necessary for the emergence of oscillations. We show that
relaxation oscillations should be expected to occur in systems with relatively
short polariton lifetime.
