Eur. Phys. J. B 83, 223-233 (2011)
https://doi.org/10.1140/epjb/e2011-20162-6

Hyperbolic diffusion in chaotic systems

¹ Department of Physical Chemistry and Technology of Polymers, Section of Physical Chemistry and Biophysics, Silesian University of Technology, 44-100 Gliwice, Poland
² Institute of Physics, University of Silesia, 40-007 Katowice, Poland

Corresponding author: ^a jerzy.luczka@us.edu.pl

Received: 4 March 2011
Revised: 23 May 2011
Published online: 23 September 2011

Abstract

We consider a deterministic process described by a discrete one-dimensional chaotic map and study its diffusive-like properties. Starting with the corresponding Frobenius-Perron equation we derive an approximate evolution equation for the probability distribution which is a partial differential equation of a hyperbolic type. Consequently, the process is correlated, non-Markovian, non-Gaussian and the information propagates with a finite velocity. This is in clear contrast to conventional diffusion processes described by a standard parabolic diffusion equation with an infinite velocity of information propagation. Our approach allows for a more complete characterisation of diffusion dynamics of deterministic systems.