A flame exhibits a limit-cycle oscillation, which is called "flame flickering" or "puffing," in a certain condition. We investigated the bifurcation structure of the flame oscillation in both simulation and experiment. We performed a two-dimensional hydrodynamic simulation by employing the flame sheet model. We reproduced the flame oscillation and investigated the parameter dependencies of the amplitude and frequency on the fuel-inlet diameter. We also constructed an experimental system, in which we could finely vary the fuel-inlet diameter, and we investigated the diameter-dependencies of the amplitude and frequency. In our simulation, we observed the hysteresis and bistability of the stationary and oscillatory states. In our experiments, we observed the switching between the stationary and oscillatory states. As fluctuations can induce the switching in the bistable system, switching observed in our experiments suggested the bistability of the two states. Therefore, we concluded that the oscillatory state appeared from the stationary state through the subcritical Andronov-Hopf bifurcation in both the simulation and experiment. The amplitude was increased and the frequency was decreased as the fuel-inlet diameter was increased. In addition, we visualized the vortex structure in our simulation and discussed the effect of the vortex on the flame dynamics.