Phenomenological and ratio bifurcations of a class of discrete time stochastic processes

Zeeman proposed a classification of stochastic dynamical systems based on the Morse classification of their invariant probability densities; the associated bifurcations are the 'phenomenological bifurcations' of L. Arnold. The classification is however not invariant under diffeomorphisms of the state space. In a recent paper we proposed an alternative classification, based on an invariant that is a ratio of joint and marginal probability density functions, which does not suffer from this defect. This classification entails the concept of what we call 'ratio bifurcations'. In this note it is shown that for a large class of dynamical systems, ratio bifurcations and phenomenological bifurcations actually coincide. Moreover, we link the ratio invariant to the transformation invariant function that Wagenmakers et al. obtained for stochastic differential equations. The results are illustrated with numerical applications to stochastic dynamical systems.