论文标题
计算圆圈的旋转子集$ \ m athbb {r}/\ mathbb {z} $在角度乘法映射下
Counting rotational subsets of the circle $\mathbb{R}/\mathbb{Z}$ under the angle multiplying map $t\mapsto dt$
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。
A rotational set is a finite subset $A$ of the unit circle $\mathbb{T}=\mathbb{R}/ \mathbb{Z}$ such that the angle-multiplying map $σ_{d}:t\mapsto dt$ maps $A$ onto itself by a cyclic permutation of its elements. Each rotational set has a geometric rotation number $p/q$. These sets were introduced by Lisa Goldberg to study the dynamics of complex polynomial maps. In this paper we provide a necessary and sufficient condition for a set to be $σ_{d}$-rotational with rotation number $p/q$. As applications of our condition, we recover two classical results and enumerate $σ_d$-rotational sets with rotation number $p/q$ that consist of a given number of orbits.
