Applying the semi-numerical approximate analytical algorithm, The property of period-doubling bifurcations in the Duffing oscillator was verified and analyzed. High order with high precision harmonic approximate analytical solution of the Duffing oscillator was obtained by incremental harmonic balance method. This verified and analyzed process have higher precision than experimental methods and do not need to compute the frequency spectrum with high frequency resolution. The results is in agreement with Feigenbaum’s theory. The theory is proved right further.