On the trajectories of CRL…LR…RCRL…LR…R orbits, their period-doubling cascades and saddle-node bifurcation cascades

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…RCRL…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.

Research highlights
► Symbolic sequences are the usual topological approach to dynamical systems.
► Permutations bear more physical information than symbolic sequences.
► Period-doubling cascade permutations associated with original sequences are obtained.
► Saddle-node cascade permutations associated with original sequences are obtained.
► Composite permutations are derived.