Composition law of cardinal order permutations

TL;DR

This paper proves theorems that establish composition laws for cardinal ordering permutations and their inverses, enabling complete understanding of point orderings in hs-periodic orbits regardless of their bifurcation origin.

Contribution

It introduces new theorems and algorithms that determine the order of points in hs-periodic orbits, applicable across various bifurcation scenarios.

Findings

Theorems for composition laws of permutations and their inverses.

Algorithms for practical application of the composition laws.

Unified understanding of point orderings in different types of periodic orbits.

Abstract

In this paper the theorems that determine composition laws for both cardinal ordering permutations and their inverses are proven. So, the relative positions of points in a hs-periodic orbit become completely known as well as in which order those points are visited. No matter how a hs-periodic orbit emerges, be it through a period doubling cascade (s=2^n) of the h-periodic orbit, or as a primary window (like the saddle-node bifurcation cascade with h=2^n), or as a secondary window (the birth of a $s-$periodic window inside the h-periodic one). Certainly, period doubling cascade orbits are particular cases with h=2 and s=2^n. Both composition laws are also shown in algorithmic way for their easy use.