Property of period-doubling bifurcations

The period-doubling bifurcation leads a T-periodic solution to a 2T-periodic solution. We develop the relation between these two periodic solutions analytically for a general parameter-dependent dynamic system. Such the relation is further confirmed by one example and shows that the 2T-periodic solution contains all the information of the T-periodic solution near the bifurcation point. Therefore we can infer the T-periodic solution from the 2T-periodic solution. Conversely, we may obtain the part of the 2T-periodic solution from the T-periodic solution. The work sheds light on the period-doubling bifurcation and chaos in general, the self-similarity of chaotic solutions in particular, forms a benchmark of numerical accuracy checking and provides new numerical schemes of period-doubling bifurcation detection.