标题： 随机现象由混沌现象近似的形式证明 显示英文标题

标题： Formal proof of the approximation of a random phenomenon by a chaotic phenomenon

摘要： 这篇文章探讨了混沌和随机性之间的微妙关系，这两个概念虽然都指看似不可预测的现象，但它们基于根本不同的原理。 混沌体现在确定性系统中，初始条件的微小变化会导致不可预测的长期行为，而随机性则涉及本质上概率性的过程，其特征是基本的不确定性。 尽管这些现象基于不同的机制，但在建模自然现象、气候预测或金融市场等不同情境中，它们可以相互作用并趋于一致。 尽管存在差异，这两种现象具有共同的特征，例如缺乏明显的秩序以及一种难以被我们长期预测的不可预测性。 通过分析混沌理论和概率论，本文旨在阐明两者的区别，并突出这两个概念在现实系统中的深层联系。 本文的目的是提出一种全面的方法，旨在证明在某些条件下，随机现象可以有效地由混沌现象来表示和近似。 通过考察这种可能性，我们试图建立连接这两个概念的理论基础，这两个概念通常被视为截然不同，但在某些情况下，它们的动力学可能证明是相似的。 显示英文摘要

摘要： This article examines the subtle relationship between chaos and randomness, two concepts that, although they refer to seemingly unpredictable phenomenon, are based on fundamentally different principles. Chaos manifests in deterministic systems where small variations in initial conditions lead to unpredictable long-term behaviors, while randomness pertains to intrinsically probabilistic processes, characterized by fundamental uncertainty. Although these phenomena are based on distinct mechanisms, they can interact and converge in contexts as varied as the modeling of natural phenomena, climate forecasts, or financial markets. Despite their differences, these two phenomena share common characteristics, such as the absence of apparent order and an unpredictability that defies our attempts at long-term prediction. Through an analysis of chaos theory and probability, this article aims to clarify the distinctions and highlight the deep connections between these two concepts in real systems. The objective of this article is to present a comprehensive approach aimed at demonstrating that, under certain conditions, a random phenomenon can be effectively represented and approximated by a chaotic phenomenon. By examining this possibility, we seek to establish the theoretical foundations that connect these two concepts, often perceived as distinct, but whose dynamics could prove to be analogous in certain contexts.