arXiv 0810.47022008-10-26nlin.CD

混沌迭代的拓扑

The topology of chaotic iterations

Jacques M. Bahi, Christophe Guyeux

摘要：

Chazan，Mi- ranker [6]和Miellou [10]在数值分析中引入了混沌迭代，而Robert [12]和Pellegrin [11]在离散动力系统框架中引入了混沌迭代。在这两种情况下，目标是获得这样的迭代收敛到一个固定的状态的条件。本文提出了一种新的观点，目的是从严格的数学意义上推导出混沌迭代具有混沌行为的条件。与文献中所研究的相反，趋同是不可取的。更确切地说，本文建立了有限集上混沌迭代概念与拓扑混沌概念之间的联系[9]，[7]，[8]。我们的动机是我们的方法的具体应用，如在计算机安全领域中使用混沌布尔迭代。事实上，混沌的概念被用于许多数据安全领域，而没有真正严格的理论基础，也没有使用允许混沌的基本属性。本文的愿望是在这一领域带来更多的数学严谨性。本文是文[3]的推广，是一项正在进行的工作.

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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