Exciton-polariton Bose-Einstein condensates (BEC) are newly emerged systems capable of showing macroscopic quantum phenomena with intrinsic open-dissipative nature. Being quasiparticles resulting from strong coupling between matter and light inside a microcavity, exciton-polaritons can be excited (pumped) by an external laser, either coherently or incoherently. Whereas the former transfers the phase of the pumping laser directly to a polariton condensate, the latter does not. In both excitation schemes, the spatial distribution of polariton density can be controlled by the geometric shape of the pumping laser, enabling the investigation of polariton dynamics with topologically non-trivial patterns. Meanwhile, exciton-polariton has spin degree of freedom inherited from photon spins, making it a candidate for realization of quantum logic gates. In our research, we investigated dynamics concerning both polariton’s spatial degree of freedom and spin degree of freedom, and the interaction between them. We found that, under the incoherent pumping scheme, for a single-component polariton condensate under an annular pump it would exhibit dynamical instability and energetic-like instability which leads to the appearance of polariton superfluidity; for a two-component polariton condensate, there exist multiple stationary steady state solutions that allow for the controlled spin state switching dynamics. Finally, the spin-orbit coupling interaction will give rise to striking phenomena such as spin waves and spontaneous vortex formation. These nonlinear effects can be well described by our unified field theory and ready to be tested in experiments.