According to four different operating modes of a BIFRED converter, a piecewise smooth linear time-varying model for pulse train (PT)-controlled BIFRED converter operating in dual discontinuous conduction modes (DCM-DCM) is established, from which time-domain waveforms and phase portraits under different loads are obtained by numerical simulations. By analyzing the variation of the charge of an output capacitor in a switching period, a one-dimensional approximate discrete mapping model for PT-controlled DCM-DCM BIFRED converter is derived, in which the multi-periodic behaviors with the variations of the load resistance and the input voltage are studied, and the stability analysis and the estimations of the load resistance ranges are performed. Research findings indicate that the dynamical behaviors depicted by the one-dimensional approximate discrete mapping model are in good agreement with those described by the piecewise smooth linear time-varying model, well revealing the complicated multi-periodic behaviors existing in the PT-controlled DCM-DCM BIFRED converter. By PSIM circuit simulations and physical circuit experimental measurements, the waveforms obtained are consistent with the results of theoretical analysis, effectively validating the feasibility of the two dynamical models.