论文标题
一种分化协变量lyapunov载体的基平均方法
An ergodic averaging method to differentiate covariant Lyapunov vectors
论文作者
论文摘要
协变量的lyapunov载体或CLV跨越了混乱的动力学系统中轨迹的扰动的扩展和收缩方向。由于有效地计算仅利用轨迹信息的算法，它们已被广泛应用于科学学科，主要用于敏感性分析和不确定性下的预测。在本文中，我们开发了一种数值方法来计算CLV沿着自己的方向的定向衍生物。与CLV的计算相似，其衍生物的当前方法是迭代的，并且类似地使用沿轨迹的混沌图的二阶导数，除了Jacobian之外。我们在超级合同的Smale-Williams电磁吸引子上验证了新方法。我们还在其他几个示例中演示了算法，包括平滑扰动的Arnold Cat Maps和Lorenz吸引子，从而获得了每个吸引子曲率的可视化。此外，我们揭示了CLV自源性与统计线性响应公式的基本联系。
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.
