Non-local Linearization of Nonlinear Differential Equations via Polyflows

TL;DR

This paper introduces a novel hierarchy-based method for approximating nonlinear differential equations with linear ones, capturing both local and global dynamics, including stability properties, with proven convergence and empirical validation.

Contribution

The paper presents a new approximation scheme using polynomial vector flows that captures local and global behaviors, including asymptotic stability, surpassing traditional Taylor methods.

Findings

The approximation scheme converges at least as well as Taylor expansions.

It effectively captures asymptotic stability in nonlinear systems.

Empirical results demonstrate strong approximation capabilities.

Abstract

Motivated by the mathematics literature on the algebraic properties of so-called polynomial vector flows, we propose a technique for approximating nonlinear differential equations by linear differential equations. Although the idea of approximating nonlinear differential equations with linear ones is not new, we propose a new approximation scheme that captures both local as well as global properties. This is achieved via a hierarchy of approximations, where the Nth degree of the hierarchy is a linear differential equation obtained by globally approximating the Nth Lie derivatives of the trajectories. We show how the proposed approximation scheme has good approximating capabilities both with theoretical results and empirical observations. In particular, we show that our approximation has convergence range at least as large as a Taylor approximation while, at the same time, being able to account for asymptotic stability (a nonlocal behavior). We also compare the proposed approach with recent and classical work in the literature.