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

Peter Giesl

doi:10.3934/dcdsb.2020315

Article Contents

2021, Volume 26, Issue 9: 4839-4865. Doi: 10.3934/dcdsb.2020315

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

Peter Giesl

Department of Mathematics, University of Sussex, Falmer, Brighton BN1 9QH, United Kingdom

Received: May 31, 2020

Revised: August 31, 2020

Early access: October 2020

Published: September 2021

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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 enables the explicit construction of a contraction metric by numerically solving this equation in [7]. In this paper we prove existence and uniqueness of the solution of the PDE and show that it defines a contraction metric.

Keywords:

Periodic orbit,
stability,
contraction metric,
converse theorem,
matrix-valued partial differential equation,
existence,
uniqueness.

Mathematics Subject Classification: Primary: 34C25, 34D20; Secondary: 37C27.

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