Grid-connected inverter system is recognized as a time-varying nonlinear system, and it has complex nonlinear behaviors in practice. However, the introduction of dead-time nonlinearity can make the nonlinear behaviors of the system more complex and harder to predict. In this paper, the proportional control single-phase grid-connected inverter with L-filter considering dead-time nonlinearity is investigated. The observation of current waveforms under fixed controller parameter and different dead-time parameters shows that the bifurcation phenomenon occurs with the increase of dead-time. According to the features of dead-time nonlinearity and the zero current clamping phenomenon caused by dead-time, an exact and complete discrete-time model is established by comprehensively considering the system from various aspects. And the theoretical analysis of bifurcation behaviors of system is carried out based on the proposed model. Moreover, the stability judgment has engineering significance. However, with the consideration of dead-time nonlinearity, analytical methods are difficult to use and the graphic methods do not have satisfactory accuracy. Therefore, a stability criterion based on equivalent duty ratio is proposed in this paper, which can accurately judge the stability boundary and provide reliable references for the parameter design of controller and dead-time.