Introduction to Volume 1

Summary

Over the last few years there has been a quite remarkable increase of activity in the study of nonlinear dynamical systems exposed to external noise. This is especially true in relation to colored noise, that is noise whose correlation time τ is nonzero. The field as it has developed stands on a firm foundation of pioneering, mostly Russian, theoretical work on white noise driven systems dating back to the early 1930s, an exemplar being the landmark 1933 paper by Pontryagin, Andronov and Vitt, of which a first-ever translation into English appears as an appendix to this volume. The early development of the field is reviewed in Chapter 1.

The recent intensification of research effort has in large part been stimulated by two distinct but closely related factors. First, it has become increasingly apparent that most (perhaps all) real physical systems need to be considered in the context of colored noise: they cannot be described adequately within the framework of white noise theory. Secondly, it is the case that, with few exceptions, physically interesting Fokker–Planck systems cannot be solved exactly when driven by colored noise. To meet this challenge, a variety of approximation schemes, carried out within several theoretical frameworks, have been developed, and much effort has been devoted to the testing of their efficacy under different conditions.