Investigating the Phase Space Dynamics of Hamiltonian Systems by the Origin-Fate Map

TL;DR

This paper explores phase space transport in a two-dimensional caldera potential using the Origin-Fate Map and Lagrangian Descriptor analysis, revealing complex dynamics, fractal structures, and manifold-guided transport mechanisms.

Contribution

It introduces the use of the Origin-Fate Map framework combined with Lagrangian Descriptors to analyze transport in a caldera potential, highlighting new insights into channel imbalance and fractal boundary structures.

Findings

Reproduction of symmetric patterns at lambda=1.0

Identification of channel imbalance and chaotic regions at smaller lambda

Discovery of fractal-like boundary structures and lobe dynamics

Abstract

We investigate phase space transport in a two-dimensional stretched caldera potential using the Origin-Fate Map (OFM) framework, complemented by Lagrangian Descriptor (LD) analysis. The caldera potential, a model for reaction dynamics with multiple exit channels, is adjusted by a stretching factor lambda that controls the directional bias of the four-saddle landscape. Several OFMs are constructed for two Poincare surfaces of section using forwards and backwards symplectic integration to assign each initial condition a channel of origin and fate. Our results reproduce the highly symmetric lambda = 1.0 patterns reported in Hillebrand et al. (Phys. Rev. E 108, 024211, 2023), and reveal, for smaller lambda, pronounced channel imbalance, figure-eight transport loops, and complex mixed-channel chaotic regions. Long-time integrations show a reduction of trapped regions with boundaries that exhibit self-similarity under deep zoom, revealing fractal-like structures. High-resolution OFMs and LD gradient maps uncover lobe dynamics and manifold structures that govern transport, showing near-perfect alignment between LD ridges and OFM boundaries.