Algebra, coalgebra, and minimization in polynomial differential equations

TL;DR

This paper introduces an algebraic and coalgebraic framework for reasoning about and minimizing polynomial differential equations using L-bisimulation, ideals, and Lie-derivatives, enabling system reduction and approximation.

Contribution

It develops a novel algebraic characterization of L-bisimilarity for polynomial ODEs and provides algorithms for system minimization and approximation.

Findings

L-bisimulation corresponds to identical solutions in polynomial ODEs.

An algebraic method using ideals characterizes L-bisimilarity.

A complete algorithm computes minimal equivalent systems via ideal chains.

Abstract

We consider reasoning and minimization in systems of polynomial ordinary differential equations (ode's). The ring of multivariate polynomials is employed as a syntax for denoting system behaviours. We endow this set with a transition system structure based on the concept of Lie-derivative, thus inducing a notion of L-bisimulation. We prove that two states (variables) are L-bisimilar if and only if they correspond to the same solution in the ode's system. We then characterize L-bisimilarity algebraically, in terms of certain ideals in the polynomial ring that are invariant under Lie-derivation. This characterization allows us to develop a complete algorithm, based on building an ascending chain of ideals, for computing the largest L-bisimulation containing all valid identities that are instances of a user-specified template. A specific largest L-bisimulation can be used to build a reduced system of ode's, equivalent to the original one, but minimal among all those obtainable by linear aggregation of the original equations. A computationally less demanding approximate reduction and linearization technique is also proposed.