DC–DC converters are widely used. They are a typical class of strongly nonlinear time-varying systems that show rich nonlinear phenomena under certain working conditions. Therefore, an in-depth study of their nonlinear phenomena, characteristics, and generation mechanism is of great practical significance for gaining a deep understanding of this kind of switching converter, revealing the essence of these nonlinear phenomena and then optimizing the design of this kind of converter. Based on the fact that most of the inductance and capacitance are fractional-order, the nonlinear dynamic characteristics of the fractional-order (FO) DCM buck–boost converter are researched in this paper. The main research work and achievements of this paper include: (1) using the predictor–corrector method of fractional calculus, which is not limited by fractional order and can directly calculate the accurate values of the inductance current and capacitor voltage of the fractional converter; the predictor–corrector model of the FO converter is established. (2) The bifurcation diagrams are obtained based on this model, and the period-doubling bifurcation and chaotic behavior of the FO buck–boost converter are analyzed. (3) The phase diagrams are obtained and verified to the point that period-doubling bifurcation occurs; then, some conclusions are drawn. The results show that under certain operating and parameters conditions, the FO buck–boost converter will appear as a bifurcation and chaotic nonlinear phenomenon. Under the condition of the same circuit parameters, the stability parameter domains of the integer-order buck–boost converter and the FO buck–boost converter are different. Compared with the integer-order converter, the parameter stability region of the FO buck–boost converter is bigger. The FO buck–boost converter is more accurate at describing the nonlinear dynamic characteristics. Furthermore, the predictor–corrector method can also be applied to other FO power converters and provides theoretical guidance for converter parameter optimization and controller design.