Universal embedding of autonomous dynamical systems into a Lotka-Volterra-like format

Abstract

We show that the ordinary differential equations (ODEs) of any deterministic autonomous dynamical system with continuous and bounded rate-field components can be embedded into a quadratic Lotka-Volterra-like form by turning to an augmented set of state variables. The key step consists in expressing the rate equations by employing the Universal Approximation procedure (borrowed from the machine learning context) with logistic sigmoid ‘activation function’. Then, by applying already established methods, the resulting ODEs are first converted into a multivariate polynomial form (also known as generalized Lotka-Volterra), and finally into the quadratic structure. Although the final system of ODEs has a dimension virtually infinite, the feasibility of such a universal embedding opens to speculations and calls for an interpretation at the physical level.