二维混沌映射的双曲性估计
Estimating hyperbolicity of chaotic bidimensional maps
摘要：
本文将Ginelli等人提出的数值方法应用于二维混沌映射，以近似相应的Oseledets分裂，即由所谓的协变李雅普诺夫向量（CLV）所构成的线性子空间的集合，并对应于李雅普诺夫谱。这些子空间是非周期点的线性不变流形的模拟，因此它们之间的角度可以用来量化一般轨道的双曲性程度;然而，在相空间的光滑变换下，这种分裂是非不变的，当坐标改变时，研究横截性的性质是有趣的，例如在不同的动力系统中研究它。为了说明这个问题上的奇里科夫-泰勒标准地图，我们比较了概率密度的横截两个不同的坐标系，这些都是连接的线性变换，变形分裂角通过相空间，也改变了概率密度的几乎为零的角度，虽然完全相切实际上是不变的。这完全是由于PDF变换定律，并强烈表明，从这种分布的任何统计推断，必须采取一般小心。
英文摘要：
We apply to bidimensional chaotic maps the numerical method proposed by Ginelli et al. to approximate the associated Oseledets splitting, i.e. the set of linear subspaces spanned by the so called covariant Lyapunov vectors (CLV) and corresponding to the Lyapunov spectrum. These subspaces are the analog of linearized invariant manifolds for non-periodic points, so the angles between them can be used to quantify the degree of hyperbolicity of generic orbits; however, being such splitting non invariant under smooth transformations of phase space, it is interesting to investigate the properties of transversality when coordinates change, e.g. to study it in distinct dynamical systems. To illustrate this issue on the Chirikov-Taylor standard map we compare the probability densities of transversality for two different coordinate systems; these are connected by a linear transformation that deforms splitting angles through phase space, changing also the probability density of almost-zero angles although complete tangencies are in fact invariant. This is completely due to the PDF transformation law and strongly suggests that any statistical inference from such distributions must be generally taken with care.
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