A Sufficient Condition for the Super-linearization of Polynomial Systems

TL;DR

This paper establishes a new sufficient condition for polynomial systems to be super-linearizable, based on the properties of a weighted dependency graph, and provides an algorithm for constructing such linearizations.

Contribution

It introduces a novel weighted dependency graph criterion for super-linearization and offers a constructive algorithm to find linearizations of polynomial systems.

Findings

The condition is that the product of edge weights along any cycle in the graph is constant.

The paper provides an explicit algorithm to obtain super-linearizations.

Demonstrates the approach with a concrete example.

Abstract

We provide in this paper a sufficient condition for a polynomial dynamical system $\dot x(t) = f(x(t))$ to be super-linearizable, i.e., to be such that all its trajectories are linear projections of the trajectories of a linear dynamical system. The condition is expressed in terms of the hereby introduced weighted dependency graph $G$, whose nodes $v_i$ correspond to variables $x_i$ and edges $v_iv_j$ have weights $\frac{\partial f_j}{\partial x_i}$. We show that if the product of the edge weights along any cycle in $G$ is a constant, then the system is super-linearizable. The proof is constructive, and we provide an algorithm to obtain super-linearizations and illustrate it on an example.