On the stability of periodic orbits for differential systems in $\mathbb{R}^n$

TL;DR

This paper introduces a new method for computing characteristic multipliers of periodic orbits in autonomous differential systems, offering an alternative to traditional variational equations, and demonstrates its application in stability analysis with examples.

Contribution

A novel approach for calculating characteristic multipliers of periodic orbits in $\,\mathbb{R}^n$, applicable when orbits are intersections of hypersurfaces, bypassing first order variational equations.

Findings

Method successfully computes characteristic multipliers in various examples.

Application to rigid body dynamics confirms stability analysis effectiveness.

Provides an alternative computational technique for periodic orbit stability.

Abstract

We consider an autonomous differential system in $\mathbb{R}^n$ with a periodic orbit and we give a new method for computing the characteristic multipliers associated to it. Our method works when the periodic orbit is given by the transversal intersection of $n-1$ codimension one hypersurfaces and is an alternative to the use of the first order variational equations. We apply it to study the stability of the periodic orbits in several examples, including a periodic solution found by Steklov studying the rigid body dynamics.