轨迹弧长揭示混沌
Trajectory arclength reveals chaos
AI总结：
轨迹弧长通过时间平均拉格朗日描述符揭示混沌特性，提供了一种简单且高效的几何混沌指示器，适用于高维系统和机器学习训练。
摘要：
在本文中，我们证明了轨迹的相空间弧长，通过时间平均的拉格朗日描述符量化，是一种鲁棒且自包含的混沌指示器。通过应用伯克霍夫的遍历分区定理，我们展示该标量函数通过其功率谱密度区分动态区域：对于规则运动，它收敛到一个delta函数，而对于混沌轨迹，谱呈现出由动态粘滞现象驱动的反幂律(1/ω)。通过这种方法，我们避免了变分方程的计算和邻近轨道的模拟，使其成为从拉格朗日描述符导出的最简单的几何混沌指示器。其计算效率使研究高维系统成为可能，并允许生成大量分类的初始条件数据集，非常适合用于训练机器学习模型。我们通过赫恩-海尔斯和费米-帕斯塔-乌拉姆系统验证了这些发现。通过将相空间的几何特性与谱分析联系起来，本文为建立拉格朗日描述符作为严格、自足框架的数学依据，用于哈密顿系统中混沌和规则性的全局分析。
英文摘要：
In this paper, we investigate the time-averaged phase space arclength of a trajectory, expressed through a normalized Lagrangian descriptor, as a trajectory-based diagnostic for distinguishing regular and chaotic dynamics. The proposed indicator, $Ω$, is computed by integrating the equations of motion together with a single scalar arclength-accumulation equation and therefore does not require variational equations, neighboring trajectories, or spatial derivatives of a Lagrangian descriptor field. Using Birkhoff's ergodic theorem together with estimates for quasiperiodic motion and correlation-based arguments for chaotic ergodic dynamics, we show that regular and chaotic trajectories exhibit different finite-time behavior despite the convergence of $Ω$ in both regimes. For quasiperiodic trajectories with Diophantine frequency vectors, deviations from the limiting average decay as $\mathcal{O}(τ^{-1})$, whereas sufficiently mixing chaotic dynamics exhibit root-mean-square fluctuations of order $τ^{-1/2}$ under suitable correlation assumptions. This difference in finite-time convergence and fluctuation scaling provides the theoretical basis for using $Ω$ to discriminate between the two dynamical regimes. We validate the proposed approach in the Hénon--Heiles and Fermi--Pasta--Ulam systems, comparing the resulting classification of initial conditions with SALI as a reference chaos indicator. The results support $Ω$ as a complementary single-trajectory diagnostic for chaos detection and as a practical source of scalar features for the analysis of trajectory ensembles, particularly in settings involving large numbers of trajectories or high-dimensional systems.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
