# American Institute of Mathematical Sciences

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

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

Received  June 2020 Revised  September 2020 Published  October 2020

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.

Citation: Peter Giesl. On a matrix-valued PDE characterizing a contraction metric for a periodic orbit. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020315
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##### References:
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