WORKING PAPERS SERIES
WP05-05
Equivalence and bifurcations of
finite order stochastic processes
Cees Diks and Florian Wagener
Equivalence and bifurcations of finite order stochastic
processes !
C. Diks and F.O.O. Wagener †
25th April 2005
Abstract
This article presents an equivalence notion of finite order stochastic processes. Local dependence mea-
sures are defined in terms of ratios of joint and marginal probability densities. The dependence measures
are classified topologically using level sets. The corresponding bifurcation theory is illustrated with
some simple examples.
1 Introduction
Bifurcation theory has been an extremely successful tool to investigate the qualitative (or structural) prop-
erties of deterministic nonlinear systems. But in many practical situations, deterministic models fit the
available data only imperfectly, and stochastic models are proposed to describe the behaviour of a given
system; the stochastic components can model genuinely random events, but they can also be introduced for
quantities of which not enough is known to describe them otherwise.
Motivated by the success of deterministic bifurcation theory, there have been several attempts to
develop bifurcation theory for stochastic processes; however, to find a natural replacement of the notion of
‘topological equivalence’ has been the main problem. For at the base of any bifurcation theory, there is a
notion of ‘form’ or ‘shape’, formalised as an equivalence relation between systems: two systems are said to
be of the same form if they are in the same equivalence class. A meaningful bifurcation theory can only be
developed if there are equivalence classes with non-empty interior; note that this presupposes a topology on
the space of systems. Elements in the interior of an equivalence class are ‘structurally stable’: if a system
parameter is changed slowly, the system will remain in the equivalence class and the form of the system
does not change. All other elements, associated to changes of form, are called ‘bifurcatin