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

Abstract

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.