On the trajectories of [formula omitted] orbits, their period-doubling cascades and saddle-node bifurcation cascades

Abstract

In this Letter, it is shown that from a two region partition of the phase space of a one-dimensional dynamical system, a p-region partition can be obtained for the CRL … LR … R orbits. That is, permutations associated with symbolic sequences are obtained. As a consequence, the trajectory in phase space is directly deduced from permutation. From this permutation other permutations associated with period-doubling and saddle-node bifurcation cascades are derived, as well as other composite permutations.