Equivalence between nonlinear dynamical systems and urn processes

• Published in: Journal of physics. A, Mathematical and theoretical Vol. 51; no. 48; pp. 485101 - 485114
• Main Authors: Brenig, Léon, Gleria, Iram, Rocha Filho, Tarcísio M, Figueiredo, Annibal, Hernández-Bermejo, Benito
• Format: Journal Article
• Language: English
• Published: IOP Publishing 30.11.2018
• Subjects: canonical forms; Lotka-Volterra systems; nonlinear dynamical systems; urn processes
• ISSN: 1751-8113; 1751-8121
• DOI: 10.1088/1751-8121/aae770

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 (2006 Discrete Mathematics and Theoretical Computer Science, AG (DMTCS Proc.) pp 59-118) 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.