We show that turbulent dynamics that arise in simulations of the three-dimensional Navier-Stokes equations in a triply periodic domain under sinusoidal forcing can be described as transient visits to the neighborhoods of unstable time-periodic solutions. Based on this description, we reduce the original system with more than ${10}^5$ degrees of freedom to a 17-node Markov chain where each node corresponds to the neighborhood of a periodic orbit. The model accurately reproduces long-term averages of the system’s observables as weighted sums over the periodic orbits.

中文翻译：

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使用周期轨道粗粒度化湍流的状态空间

我们表明，在正弦强迫下三重周期域中三维纳维-斯托克斯方程的模拟中出现的湍流动力学可以描述为对不稳定时间周期解的邻域的瞬态访问。基于这个描述，我们将原始系统减少了超过${10}^5$17 节点马尔可夫链的自由度，其中每个节点对应于周期轨道的邻域。该模型准确地再现了系统可观测值的长期平均值，作为周期轨道上的加权总和。