Reducing nonlinear dynamical systems to canonical forms

A global framework for treating nonlinear differential dynamical systems is presented. It rests on the fact that most systems can be transformed into the quasi-polynomial format. Any system in this format belongs to an infinite equivalence class characterized by two canonical forms, the Lotka–Volterra (LV) and the monomial systems. Both forms allow for finding total or partial integrability conditions, invariants and dimension reductions of the original systems. The LV form also provides Lyapunov functions and systematic tools for stability analysis. An abstract Lie algebra is shown to underlie the whole formalism. This abstract algebra can be expressed in several realizations among which are the bosonic creation–destruction operators. One of these representations allows one to obtain the analytic form of the general coefficient of the Taylor series representing the solution of the original system. This generates a new class of special functions that are solutions of these nonlinear dynamical systems. From the monomial canonical form, one can prove an equivalence relationship between urn processes and dynamical systems. This establishes a new link between nonlinear dynamics and stochastic processes. This article is part of the theme issue ‘Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 1)’.