论文标题
流动在表面上的正面和负面行为的依赖性
Dependency of the positive and negative long-time behaviors of flows on surfaces
论文作者
论文摘要
储层计算是预测湍流的有力工具，其简单的架构具有处理大型系统的计算效率。然而，其实现通常需要完整的状态向量测量和系统非线性知识。我们使用非线性投影函数将系统测量扩展到高维空间，然后将其输入到储层中以获得预测。我们展示了这种储层计算网络在时空混沌系统上的应用，该系统模拟了湍流的若干特征。我们表明，使用径向基函数作为非线性投影器，即使只有部分观测并且不知道控制方程，也能稳健地捕捉复杂的系统非线性。最后，我们表明，当测量稀疏、不完整且带有噪声，甚至控制方程变得不准确时，我们的网络仍然可以产生相当准确的预测，从而为实际湍流系统的无模型预测铺平了道路。
Long-time behavior is one of the most fundamental properties in dynamical systems. The limit behaviors of flows on surfaces are captured by the Poincaré-Bendixson theorem using the $ω$-limit sets. This paper demonstrates that the positive and negative long-time behaviors are not independent. In fact, we show the dependence between the $ω$-limit sets and the $α$-limit sets of points of flows on surfaces, which partially generalizes the Poincaré-Bendixson theorem. Applying the dependency result to solve what kinds of the $ω$-limit sets appear in the area-preserving (or, more generally, non-wandering) flows on compact surfaces, we show that the $ω$-limit set of any non-closed orbit of such a flow with arbitrarily many singular points on a compact surface is either a subset of singular points or a locally dense Q-set. Moreover, we show the wildness of surgeries to add totally disconnected singular points and the tameness of those to add finitely many singular points for flows on surfaces.
