SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 2, Page 763-787, January 2020.
Tobasco, Goluskin, and Doering [https://doi.org/10.1016/j.physleta.2017.12.023 Phys. Lett. A, 382 (2018), pp. 382--386}] 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.

中文翻译：

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用多项式优化寻找极值周期轨道，并将其应用于剪切流的九模式模型

SIAM应用动力系统杂志，第19卷，第2期，第763-787页，2020年1月。
Tobasco，Goluskin和Doering [https://doi.org/10.1016/j.physleta.2017.12.023 Phys。来吧 [A，382（2018），382--386页]最近建议，可以使用函数的子级集来定位优化某个可观测对象的无限时平均值的ODE系统的轨迹，这些子级集使用所谓的辅助功能。在本文中，我们证明了这种想法是可行的，并允许计算多项式ODE系统的极不稳定周期轨道（UPO）。首先，我们证明多项式优化可以保证产生辅助函数，这些辅助函数在时间平均上产生接近锐利的边界，这是精确定位极值轨道所必需的。其次，我们表明可以通过直接非线性优化有效地计算相关子级集中的点。这些点为UPO计算提供了良好的初始条件。作为概念验证，然后将这些方法与单次射击的Newton-Raphson算法结合使用，以研究正弦强迫剪切流的9维模型的极值UPO。我们发现了三个以前未知的UPO族，其中之一对于雷诺数大约在81.24到125之间的同时相对于层状状态最小化了平均能量耗散率并最大化了平均摄动能量。