Coupled one-dimensional dynamical systems

TL;DR

This paper introduces a class of coupled one-dimensional dynamical systems with complex behaviors, providing definitions, visual examples, and a JavaScript tool for further exploration, but lacks formal mathematical results.

Contribution

It defines a new class of coupled 1D systems and offers visualizations and tools for studying their complex behaviors, without formal proofs.

Findings

Examples of asymptotic limit sets shown visually

A JavaScript tool for exploring the systems' dynamics

Highlighting the complexity of coupled 1D systems

Abstract

This note aims to bring attention to a simple class of discrete dynamical systems exhibiting some complex behaviour. Each of these systems is defined as a self-mapping of the unit square and is obtained by coupling two families of self-mappings of the unit interval. There is no real mathematics to be found here (in the sense of results stated and proved) and in fact there is an almost complete lack of precise statements. The only thing on offer is the definition of the mappings and a few nice pictures showing examples of their asymptotic limit sets. There is a JavaScript program available, accessible at www.math.uni-bielefeld.de/~preston/iterates.html, which can be used to `discover' more about these mappings. The program might prove to be helpful for anyone interested in doing this.