On the dynamics of skew tent maps

TL;DR

This paper provides a detailed analysis of the dynamics of skew tent maps, explicitly dividing parameter space into regions, identifying attractors, and characterizing their nature as periodic or chaotic.

Contribution

It offers an explicit classification of the attractors and dynamics of skew tent maps across different parameter regions, including the existence of invariant chaotic sets.

Findings

Attractors are either periodic or chaotic intervals.

Chaotic attractors can be single intervals or unions of intervals.

Periodic attractors can have any period.

Abstract

In this paper we give an elementary treatment of the dynamics of skew tent maps. We divide the two-parameter space into six regions. Two of these regions are further subdivided into infinitely many regions. All of the regions are given explicitly. We find the attractor in each subregion, determine whether the attractor is a periodic orbit or is chaotic, and also determine the asymptotic fate of every point. We find that when the attractor is chaotic, it is either a single interval or the disjoint union of a finite number of intervals; when it is a periodic orbit, all periods are possible. Sometimes, besides the attractor, there exists an invariant chaotic Cantor set.