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

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