Property of period-doubling bifurcation cascades of discrete dynamical systems

The period-doubling bifurcation process is one of the routes to chaos. During this process, when the system parameter reaches the bifurcation points, the periodic orbits lose their stability and double their periods. It is found that the periodic orbits born in this process exhibit an intimate relationship. On one hand, the orbits with relatively small periods can be approximately extracted from the orbit with a large period; on the other hand, the orbits with large periods can be approximately constructed by the orbit with relatively small period. Furthermore, our analytical results strongly suggest that the unstable periodic orbits originating from the period-doubling bifurcation process should play a big role in the ensuing chaos, at least at its early stage.