With the popularity of distributed generations (DGs), electric energy routers (EER) integrating DGs have been widely used in low-voltage distribution network (LVDN). A series-parallel architecture EER (SPA-EER) for LVDN has been proposed to make up for shortcomings of series architecture EER (SA-EER) in energy flexible operation. However, when some parameters in the SPA-EER system change beyond a certain range, the system will present low- or high-frequency bifurcation behaviors, resulting in system oscillation and instability. To analyze and suppress bifurcation behaviors in the SPA-EER system, the discrete models of series and parallel converters are respectively established to obtain the operational stability domains of above parameters. Then, a new bifurcation behavior suppression method is proposed to enhance these stability domains, in which a second-order generalized integrator (SOGI) is introduced into the traditional time-delayed feedback control (TDFC), namely SOGI-TDFC, to compensate for the shortcoming of traditional TDFC. Finally, the simulation results verify the correctness of the theoretical analysis and the proposed control method.