Equivalence between nonlinear dynamical systems and urn processes

TL;DR

This paper establishes a connection between a broad class of deterministic quasi-polynomial dynamical systems and stochastic balanced urn processes, bridging mathematical modeling and combinatorics.

Contribution

It extends previous work by showing an equivalence between general dynamical systems and urn processes, broadening the scope of the existing theorem.

Findings

Demonstrates the equivalence between quasi-polynomial differential systems and urn processes.

Extends Flajolet et al.'s theorem to more general dynamical systems.

Provides a new perspective linking dynamical systems and combinatorial probability.

Abstract

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.