Deterministic Brownian motion generated from differential delay equations

TL;DR

This paper demonstrates how deterministic differential delay equations can produce Brownian-like motion through chaotic solutions, revealing Gaussian-like densities and statistical properties similar to classical Brownian motion.

Contribution

It analytically and numerically investigates the probabilistic behavior of solutions to a simple differential delay equation, linking chaos to stochastic-like motion.

Findings

Solutions exhibit Gaussian-like density over time

Chaotic solutions can generate Brownian-like motion

Probabilistic properties may be universal for similar chaotic systems

Abstract

This paper addresses the question of how Brownian-like motion can arise from the solution of a deterministic differential delay equation. To study this we analytically study the bifurcation properties of an apparently simple differential delay equation and then numerically investigate the probabilistic properties of chaotic solutions of the same equation. Our results show that solutions of the deterministic equation with randomly selected initial conditions display a Gaussian-like density for long time, but the densities are supported on an interval of finite measure. Using these chaotic solutions as velocities, we are able to produce Brownian-like motions, which show statistical properties akin to those of a classical Brownian motion over both short and long time scales. Several conjectures are formulated for the probabilistic properties of the solution of the differential delay equation. Numerical studies suggest that these conjectures could be "universal" for similar types of "chaotic" dynamics, but we have been unable to prove this.