微分协变李雅普诺夫向量的遍历平均法
An ergodic averaging method to differentiate covariant Lyapunov vectors
摘要：
协变李雅普诺夫向量（CLV）跨越混沌动力系统中沿轨迹的扰动的扩张和收缩方向。由于有效的算法来计算它们，只利用轨迹信息，它们已被广泛应用于各科学学科，主要用于敏感性分析和不确定性下的预测。在本文中，我们发展了一种数值方法来计算CLV沿自己的方向的方向导数。类似于CLV的计算，其导数的本方法是迭代的，并且类似地除了雅可比矩阵之外还使用混沌映射沿轨迹的二阶导数。我们验证了一个超收缩Smal-Williams螺线管吸引子的新方法。我们还演示了其他几个例子，包括光滑扰动的Arnold猫地图，和洛伦兹吸引子，获得可视化的曲率的每个吸引子的算法。此外，我们揭示了一个基本的连接CLV自导数的统计线性响应公式。
英文摘要：
Covariant Lyapunov vectors or CLVs span the expanding and contracting directions of perturbations along trajectories in a chaotic dynamical system. Due to efficient algorithms to compute them that only utilize trajectory information, they have been widely applied across scientific disciplines, principally for sensitivity analysis and predictions under uncertainty. In this paper, we develop a numerical method to compute the directional derivatives of CLVs along their own directions. Similar to the computation of CLVs, the present method for their derivatives is iterative and analogously uses the second-order derivative of the chaotic map along trajectories, in addition to the Jacobian. We validate the new method on a super-contracting Smale-Williams Solenoid attractor. We also demonstrate the algorithm on several other examples including smoothly perturbed Arnold Cat maps, and the Lorenz attractor, obtaining visualizations of the curvature of each attractor. Furthermore, we reveal a fundamental connection of the CLV self-derivatives with a statistical linear response formula.
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