Deterministic Brownian motion generated from differential delay equations

摘要

This paper addresses the question of how Brownian-like motion can arise fromthe solution of a deterministic differential delay equation. To study this weanalytically study the bifurcation properties of an apparently simpledifferential delay equation and then numerically investigate the probabilisticproperties of chaotic solutions of the same equation. Our results show thatsolutions of the deterministic equation with randomly selected initialconditions display a Gaussian-like density for long time, but the densities aresupported on an interval of finite measure. Using these chaotic solutions asvelocities, we are able to produce Brownian-like motions, which showstatistical properties akin to those of a classical Brownian motion over bothshort and long time scales. Several conjectures are formulated for theprobabilistic properties of the solution of the differential delay equation.Numerical studies suggest that these conjectures could be "universal" forsimilar types of "chaotic" dynamics, but we have been unable to prove this.