Classification of the Second Minimal Odd Periodic Orbits in the Sharkovskii Ordering

TL;DR

This paper fully classifies second minimal odd periodic orbits in continuous real maps, revealing there are 4k-3 types for each orbit length, characterized by unique permutations and transition graphs.

Contribution

It provides the first complete classification of second minimal odd periodic orbits in the Sharkovskii ordering for continuous real maps.

Findings

Identifies 4k-3 types of second minimal orbits for each odd period

Characterizes each orbit by a unique cyclic permutation

Describes transition graphs with high accuracy

Abstract

This paper presents full classification of second minimal odd periodic orbits of a continuous endomorphisms on the real line. A $(2k+1)$-periodic orbit ($k\geq 3$) is called second minimal for the map $f$, if $2k-1$ is a minimal period of $f$ in the Sharkovskii ordering. We prove that there are $4k-3$ types of second minimal $(2k+1)$-orbits, each characterized with unique cyclic permutation and directed graph of transitions with accuracy up to inverses.