# Finding extremal periodic orbits with polynomial optimisation, with application to a nine-mode model of shear flow

## TL;DR

This paper develops a polynomial optimization approach to find extremal unstable periodic orbits in polynomial ODE systems, demonstrated on a shear flow model, discovering new UPOs and providing a method for accurate localization.

## Contribution

It introduces a method combining polynomial optimization and auxiliary functions to accurately localize extremal UPOs in polynomial ODE systems, validated on a shear flow model.

## Key findings

- Discovered three new families of UPOs in shear flow model.
- Proved polynomial optimization yields near-sharp bounds on time averages.
- Provided an efficient way to compute initial conditions for UPOs.

## Abstract

Tobasco et al. [Physics Letters A, 382:382-386, 2018; see https://doi.org/10.1016/j.physleta.2017.12.023] recently suggested that trajectories of ODE systems that optimize the infinite-time average of a certain observable can be localized using sublevel sets of a function that arise when bounding such averages using so-called auxiliary functions. In this paper we demonstrate that this idea is viable and allows for the computation of extremal unstable periodic orbits (UPOs) for polynomial ODE systems. First, we prove that polynomial optimization is guaranteed to produce auxiliary functions that yield near-sharp bounds on time averages, which is required in order to localize the extremal orbit accurately. Second, we show that points inside the relevant sublevel sets can be computed efficiently through direct nonlinear optimization. Such points provide good initial conditions for UPO computations. As a proof of concept, we then combine these methods with a single-shooting Newton-Raphson algorithm to study extremal UPOs for a nine-dimensional model of sinusoidally forced shear flow. We discover three previously unknown families of UPOs, one of which simultaneously minimizes the mean energy dissipation rate and maximizes the mean perturbation energy relative to the laminar state for Reynolds numbers approximately between 81.24 and 125.