Quasiperiodicity: Rotation Numbers
Quasiperiodicity: Rotation Numbers
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准周期性：旋转数
DOI：
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发表时间：
2016
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影响因子：
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通讯作者：
J. Yorke
中科院分区：
文献类型：
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作者：
Suddhasattwa Das;Yoshitaka Saiki;E. Sander;J. Yorke
如果变量的变化将环面上的映射转换为在环面的每个坐标下的纯旋转，则称环面上的映射为“准周期”。我们发展了一种寻找变量变化的数值方法，这种方法可以有效地确定变量变化的平滑程度(即可微性)，即使在具有较大非线性的情况下也是如此。我们的方法依赖于对遍历平均极限的快速而准确的估计。我们考虑的不是沿N点轨迹给点分配相同权重的均匀平均，而是权重分布不均匀的平均，它对轨迹的早、晚期点的权重远小于中点N/2附近的点。我们以一维拟周期映射为例，证明了在f充分可微的情况下，我们的加权平均收敛速度远远快于通常的O(1/N)。我们用这种方法有效地数值计算了准周期系统的转动数、不变密度、共轭，并证明了变量的变化是(实)解析的。
A map on a torus is called “quasiperiodic” if there is a change of variables which converts it into a pure rotation in each coordinate of the torus. We develop a numerical method for finding this change of variables, a method that can be used effectively to determine how smooth (i.e., differentiable) the change of variables is, even in cases with large nonlinearities. Our method relies on fast and accurate estimates of limits of ergodic averages. Instead of uniform averages that assign equal weights to points along the trajectory of N points, we consider averages with a non-uniform distribution of weights, weighing the early and late points of the trajectory much less than those near the midpoint N∕2. We provide a one-dimensional quasiperiodic map as an example and show that our weighted averages converge far faster than the usual rate of O(1∕N), provided f is sufficiently differentiable. We use this method to efficiently numerically compute rotation numbers, invariant densities, conjugacies of quasiperiodic systems, and to provide evidence that the changes of variables are (real) analytic.