We study nonlinear transport phenomena in a high-mobility bilayer system with two closely spaced populated electronic subbands in a perpendicular magnetic field. For a moderate direct current excitation, we observe zero-differential-resistance states with a characteristic 1/B periodicity. We investigate, both experimentally and theoretically, the Hall-induced resistance oscillations which modulate the high-frequency magnetointersubband oscillations in our system if we increase the current. We also observe and describe the influence of direct current on the magnetoresistance in the presence of microwave irradiation.