  An equivalence is shown between a large class of deterministic dynamical
systems and a class of stochastic processes, the balanced urn processes. These
dynamical systems are governed by quasi-polynomial differential systems that
are widely used in mathematical modeling while urn processes are actively
studied in combinatorics and probability theory. The presented equivalence
extends a theorem by Flajolet et al. (Flajolet, Dumas and Puyhaubert Discr.
Math. Theor. Comp. Sc. AG - 2006, DMTCS Proceedings) already establishing an
isomorphism between urn processes and a particular class of differential
systems with monomial vector fields. The present result is based on the fact
that such monomial differential systems are canonical forms for more general
dynamical systems.
