Deterministic Brownian Motion: The Effects of Perturbing a Dynamical System by a Chaotic Semi-Dynamical System

TL;DR

This paper reviews and extends limit theorems for chaotic deterministic systems, demonstrating how Brownian motion and Ornstein-Uhlenbeck processes can emerge from purely deterministic dynamics, suggesting chaos can mimic noise in data.

Contribution

It introduces new limit theorems for chaotic semi-dynamical systems and shows how classical stochastic processes can arise deterministically.

Findings

Brownian motion-like behavior can be derived from deterministic chaos.

Ornstein-Uhlenbeck processes can be realized within deterministic systems.

Chaotic dynamics may be mistaken for noise in experimental data.

Abstract

Here we review and extend central limit theorems for highly chaotic but deterministic semi-dynamical discrete time systems. We then apply these results show how Brownian motion-like results are recovered, and how an Ornstein-Uhlenbeck process results within a totally deterministic framework. These results illustrate that the contamination of experimental data by "noise" may, under certain circumstances, be alternately interpreted as the signature of an underlying chaotic process.