Choosing the current of the front inductor as control object, the discrete time model of single-ended primary-inductor converter (SEPIC) converter working in the discontinuous conduction mode (DCM) is established in this paper, and the stability of the fixed point is analyzed. By studying the phase portrait, power spectrum and bifurcation diagram of the circuitry, a special period-doubling bifurcation phenomenon is found in this circuitry. The circuit experiences period, double period, quadruple period and then double period, quadruple period, finally enters into chaotic condition. The experimental results accord with the simulation results, which confirms that the special period-doubling bifurcation exists in the SEPIC converter working in the DCM.