A bifurcation theory for a class of discrete time Markovian stochastic systems (Q1000768)

A bifurcation theory of smooth stochastic dynamical systems \[ X_{t+1} = g(X_t,\varepsilon_t),\quad t \in \mathbb{N} \] with everywhere positive transition densities is presented. The ``dependence ratio'' of joint and marginal stationary probability densities is used to classify its behavior with respect to possible bifurcations. This ``dependence ratio'' is a geometric invariant of the system. The authors introduce an equivalence relation defined on those ratios in order to obtain a bifurcation theory for which, in the compact case, the set of stable (non-bifurcating) systems is open and dense. Two simple, but illustrative examples support their theory. An appendix states the mathematical proof-steps of their major propositions.