Journal of Applied Mathematics and Stochastic Analysis
Volume 2004 (2004), Issue 2, Pages 137-141

A description of stochastic systems using chaotic maps

Abraham Boyarsky and Paweł Góra

Department of Mathematics and Statistics, Concordia University, 7141 Sherbrooke Street West, Montreal H4B 1R6, Québec, Canada

Received 28 August 2003; Revised 4 February 2004

Copyright © 2004 Abraham Boyarsky and Paweł Góra. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Let ρ(x,t) denote a family of probability density functions parameterized by time t. We show the existence of a family {τ1:t>0} of deterministic nonlinear (chaotic) point transformations whose invariant probability density functions are precisely ρ(x,t). In particular, we are interested in the densities that arise from the diffusions. We derive a partial differential equation whose solution yields the family of chaotic maps whose density functions are precisely those of the diffusion.