Random Dynamical Systems

Vitor Araujo

Published 2006-08-07Version 1

The concept of random dynamical system is a comparatively recent development combining ideas and methods from the well developed areas of probability theory and dynamical systems. Due to our inaccurate knowledge of the particular physical system or due to computational or theoretical limitations (lack of sufficient computational power, inefficient algorithms or insufficiently developed mathematical or physical theory, for example), the mathematical models never correspond exactly to the phenomenon they are meant to model. Moreover when considering practical systems we cannot avoid either external noise or measurement or inaccuracy errors, so every realistic mathematical model should allow for small errors along orbits not to disturb too much the long term behavior. To be able to cope with unavoidable uncertainty about the ``correct'' parameter values, observed initial states and even the specific mathematical formulation involved, we let randomness be embedded within the model to begin with. We present the most basic classes of models in what follows, then define the general concept and present some developments and examples of applications.