用于可视化旋转动力学的角映射
Angular maps for visualizing rotational dynamics
- Bielefeld University（比勒费尔德大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出角映射作为数值工具，通过变分方程的角谱可视化离散与连续动力系统中的旋转动力学，并开发高效算法，在多个系统中验证其有效性。
AI中文摘要：
我们开发并分析了角映射，将其作为一种数值工具，用于在离散和连续时间下可视化和检测非线性、非自治动力系统中的旋转动力学。角映射为每个初始点分配沿相应轨迹的变分方程的角谱。该谱描述了由线性化动力学传输的子空间的长期平均旋转，以最大主角度度量。基于角谱理论，我们开发了基于前向和后向子空间迭代的高效数值算法。我们证明了这些算法渐近地为在前向或后向时间中占主导的子空间提供角谱值。对Hénon映射、一个平面流和Lorenz系统的应用说明了角映射如何揭示相空间中的旋转特征。对于连续时间系统，我们证明了当步长趋于零时，精确时间步映射的适当重标度角谱在Hausdorff度量下收敛到连续角谱。对于自治系统，我们还证明了一种简化算法的合理性，该算法使用连续的轨迹点来近似与流方向相关的角范围。
英文摘要：
We develop and analyze angular maps as a numerical tool for visualizing and detecting rotational dynamics in nonlinear, nonautonomous dynamical systems in discrete and continuous time. An angular map assigns to each initial point the angular spectrum of the variational equation along the corresponding trajectory. This spectrum describes the long-time average rotation of subspaces transported by the linearized dynamics, measured by maximal principal angles. Building on the theory of angular spectra, we develop efficient numerical algorithms based on forward and backward subspace iteration. We prove that these algorithms asymptotically provide angular spectral values for subspaces that are dominant in either forward or backward time. Applications to Hénon maps, a planar flow, and the Lorenz system illustrate how angular maps reveal rotational features across phase space. For continuous-time systems, we prove that suitably rescaled angular spectra of exact time-step maps converge to the continuous angular spectrum in the Hausdorff metric as the step size tends to zero. For autonomous systems, we also justify a simplified algorithm that uses successive trajectory points to approximate the angular range associated with the flow direction.
