The topology of chaotic iterations

TL;DR

This paper investigates the conditions under which chaotic iterations on finite sets exhibit true chaotic behavior, establishing a rigorous topological chaos framework to support applications like data security.

Contribution

It introduces a rigorous topological chaos framework for chaotic iterations, linking them to chaos theory and enhancing their theoretical foundation for security applications.

Findings

Established conditions for chaos in finite set iterations

Linked chaotic iterations to topological chaos concepts

Provided mathematical rigor for chaos in data security contexts

Abstract

Chaotic iterations have been introduced on the one hand by Chazan, Mi- ranker [6] and Miellou [10] in a numerical analysis context, and on the other hand by Robert [12] and Pellegrin [11] in the discrete dynamical systems frame- work. In both cases, the objective was to derive conditions of convergence of such iterations to a fixed state. In this paper, a new point of view is presented, the goal here is to derive conditions under which chaotic iterations admit a chaotic behaviour in a rigorous mathematical sense. Contrary to what has been studied in the literature, convergence is not desired. More precisely, we establish in this paper a link between the concept of chaotic iterations on a finite set and the notion of topological chaos [9], [7], [8]. We are motivated by concrete applications of our approach, such as the use of chaotic boolean iterations in the computer security field. Indeed, the concept of chaos is used in many areas of data security without real rigorous theoretical foundations, and without using the fundamental properties that allow chaos. The wish of this paper is to bring a bit more mathematical rigour in this field. This paper is an extension of[3], and a work in progress.