On a matrix-valued PDE characterizing a contraction metric for a periodic orbit

TL;DR

This paper characterizes a contraction metric for a periodic orbit as a matrix-valued solution to a linear PDE, providing a foundation for explicit numerical construction and analysis of stability.

Contribution

It introduces a PDE-based characterization of contraction metrics for periodic orbits, proving existence and uniqueness of solutions.

Findings

Proves existence and uniqueness of the PDE solution.

Shows the PDE defines a valid contraction metric.

Lays groundwork for numerical construction of contraction metrics.

Abstract

The stability and the basin of attraction of a periodic orbit can be determined using a contraction metric, i.e., a Riemannian metric with respect to which adjacent solutions contract. A contraction metric does not require knowledge of the position of the periodic orbit and is robust to perturbations. In this paper we characterize such a Riemannian contraction metric as matrix-valued solution of a linear first-order Partial Differential Equation. This will enable the explicit construction of a contraction metric by numerically solving this equation in future work. In this paper we prove existence and uniqueness of the solution of the PDE and show that it defines a contraction metric.