Algebraic Test for Asymptotic Stability of Periodic Orbits for Polynomial Systems

TL;DR

This paper introduces an algebraic test based on contraction principles to determine the existence and asymptotic stability of periodic orbits in polynomial dynamical systems, providing a practical computational approach.

Contribution

It presents a novel algebraic method for testing stability of periodic orbits in polynomial systems, extending contraction-based techniques with a numerically feasible algorithm.

Findings

The algebraic test successfully identifies stable periodic orbits in polynomial systems.

The method is demonstrated through a numerical example.

The approach offers a practical tool for stability analysis in nonlinear dynamics.

Abstract

We will address the problem of determining the existence and asymptotic stability of a non-trivial periodic orbit in dynamical systems described by polynomial vector fields. To this end, we will lean upon the celebrated results of Borg, Olech and Hartman and newer results of Giesl, who all employ the concept of contraction for this purpose. Importantly, we formulate a numerically tractable algebraic test. The developed algorithm is illustrated in a numerical example.