We studied optical response of microcavity non-equilibrium exciton-polariton Bose-Einstein condensate with saturable nonlinearity under simultaneous resonant and non-resonant pumping. We demonstrated the emergence of multistabile behavior due to the saturation of the excitonic absorption. Stable periodic Rabi-type oscillations of the excitonic and photonic condensate components in the regime of the stationary pump and their transition to the chaotic dynamics through the cascade of Hopf bifurcations by tuning of the electrical pump are revealed.