We present a new method for generating robust guesses for unstable periodic orbits (UPOs) by post-processing turbulent data using dynamic mode decomposition (DMD). The approach relies on the identification of near-neutral, repeated harmonics in the DMD eigenvalue spectrum from which both an estimate for the period of a nearby UPO and a guess for the velocity field can be constructed. In this way, the signature of a UPO can be identified in a short time series without the need for a near recurrence to occur, which is a considerable drawback to recurrent flow analysis, the current state of the art. We first demonstrate the method by applying it to a known (simple) UPO and find that the period can be reliably extracted even for time windows of length one quarter of the full period. We then turn to a long turbulent trajectory, sliding an observation window through the time series and performing many DMD computations. Our approach yields many more converged periodic orbits (including multiple new solutions) than a standard recurrent flow analysis of the same data. Furthermore, it also yields converged UPOs at points where the recurrent flow analysis flagged a near recurrence but the Newton solver did not converge, suggesting that the new approach can be used alongside the old to generate improved initial guesses. Finally, we discuss some heuristics on what constitutes a ‘good’ time window for the DMD to identify a UPO.

中文翻译：

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用动态模式分解寻找周期轨道的湍流

我们提出了一种通过使用动态模式分解 (DMD) 对湍流数据进行后处理来生成不稳定周期轨道 (UPO) 稳健猜测的新方法。该方法依赖于对 DMD 特征值谱中近中性、重复谐波的识别，从中可以构建对附近 UPO 周期的估计和对速度场的猜测。通过这种方式，可以在短时间序列中识别 UPO 的特征，而无需发生近乎复发，这对于当前的现有技术状态的循环流分析来说是一个相当大的缺点。我们首先通过将其应用于已知（简单）UPO 来演示该方法，并发现即使对于长度为整个周期四分之一的时间窗口，也可以可靠地提取该周期。然后我们转向一个漫长的湍流轨迹，通过时间序列滑动观察窗口并执行许多 DMD 计算。我们的方法比对相同数据的标准循环流分析产生更多的收敛周期轨道（包括多个新解决方案）。此外，它还在递归流分析标记接近递归但牛顿求解器不收敛的点处产生收敛的 UPO，这表明新方法可以与旧方法一起使用以生成改进的初始猜测。最后，我们讨论了一些关于什么构成 DMD 识别 UPO 的“良好”时间窗口的启发式方法。它还在循环流分析标记接近递归但牛顿求解器未收敛的点处产生收敛的 UPO，这表明新方法可以与旧方法一起使用以生成改进的初始猜测。最后，我们讨论了一些关于什么构成 DMD 识别 UPO 的“良好”时间窗口的启发式方法。它还在循环流分析标记接近递归但牛顿求解器未收敛的点处产生收敛的 UPO，这表明新方法可以与旧方法一起使用以生成改进的初始猜测。最后，我们讨论了一些关于什么构成 DMD 识别 UPO 的“良好”时间窗口的启发式方法。