Projecting dynamical systems via a support bound

TL;DR

This paper develops a bound for the minimal differential equation of a coordinate in polynomial dynamical systems and introduces an efficient algorithm for its computation, outperforming existing software.

Contribution

It provides a sharp bound on the Newton polytope for the minimal equation and presents a novel evaluation-interpolation algorithm for differential elimination.

Findings

The bound is sharp when d ≤ D or the model is planar.

The algorithm can handle problems beyond current software capabilities.

Implementation results demonstrate improved performance in computing minimal equations.

Abstract

For a polynomial dynamical system, we study the problem of computing the minimal differential equation satisfied by a chosen coordinate (in other words, projecting the system on the coordinate). This problem can be viewed as a special case of the general elimination problem for systems of differential equations and appears in applications to modeling and control. We give a bound for the Newton polytope of such minimal equation. Our bound depends on the dimension of the model and the degrees $d$ and $D$ of the polynomials defining the dynamics of the chosen coordinate and the remaining coordinates, respectively. We show that our bound is sharp if $d \leqslant D$ or the model is planar. We further use this bound to design an algorithm for computing the minimal equation following the evaluation-interpolation paradigm. We demonstrate that our implementation of the algorithm can tackle problems which are out of reach for the state-of-the-art software for differential elimination.