AI 中文总结
本研究提出多映射对比框架，结合定制正则变换的两类新型截面，解决庞加莱截面的几何畸变问题，为多自由度系统动力学表征提供无畸变诊断工具。
AI 中文摘要
非线性哈密顿系统的全局相空间结构通常采用庞加莱截面可视化，但刚性选择的截面超平面常引入几何畸变和坐标伪影，掩盖或截断基本不变结构。本文针对平面弹性摆开展多映射分析，系统克服这些视觉与结构局限。我们构建了利用倒置相空间映射的对比框架，解决混沌边界附近不变曲线的密集堆积问题，发现标准视图中隐藏的明显分界线轨迹。利用系统的垂直对称轴，我们推导两类新型定制正则变换，分别使截面条件匹配底层力场和不变轨迹。研究表明，力线截面平衡了平衡点附近的相空间密度表示，揭示了曲率驱动的类六边形边界变形；轨迹对齐截面则将高度弯曲的不变流形展开为规则网格。结合倒置映射，这种基于轨迹的方法可作为结构坐标缩放，最小化局部度量畸变，将精细的高阶卫星岛直接移至焦点中心。结果显示，仅依赖单个截面不足以捕捉复杂非线性动力学，使用至少两个正交、符合场特性的截面，为表征多自由度系统的结构稳定性、共振链和全局输运壁垒提供了更优的无畸变诊断工具。
英文摘要
The global phase-space organization of non-linear Hamiltonian systems is traditionally visualized using Poincaré sections. However, rigid choices of sectioning hyperplanes often introduce geometric distortions and coordinate artifacts that obscure or clip fundamental invariant structures. Here, we present a multi-mapping analysis of the planar elastic pendulum to systematically overcome these visual and structural limitations. We implement a comparative framework utilizing inverted phase-space mappings that resolve the dense packing of invariant curves near chaotic boundaries, uncovering an apparent separatrix trajectory hidden in standard views. Leveraging the system's vertical symmetry axis, we derive two novel classes of customized canonical transformations that align the sectioning condition with the underlying force field and invariant trajectories, respectively. We demonstrate that the force-line section balances phase-space density representation near equilibrium and exposes a curvature-driven, hexagon-like boundary deformation. Concurrently, the trajectory-aligned section unrolls highly curved invariant manifolds into a regular grid. When combined with inverted mapping, this trajectory-based approach acts as a structural coordinate zoom, minimizing local metric distortions and shifting delicate, higher-order satellite islands directly into the focal center. Our results demonstrate that relying on a single slice is insufficient to capture complex non-linear dynamics; instead, utilizing at least two orthogonal, field-conforming sections provides a superior, distortion-free diagnostic tool for characterizing structural stability, resonance chains, and global transport barriers in multi-degree-of-freedom systems.
Comments27 pages, 15 figures
