A Framework for Intrinsic Poincar\'e Sections and Phase-Space Manifold Visualization: A Case Study of the Planar Elastic Pendulum

Abstract

The global phase-space organization of non-linear Hamiltonian systems is traditionally visualized using Poincar\'e 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.