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

TL;DR

This work tackles how phase-space transport in a two-degree-of-freedom caldera potential organizes itself when multiple exit channels are present. It combines the Origin--Fate Map (OFM) framework with Lagrangian Descriptors (LDs) to classify transport outcomes across a tunable stretching parameter $\lambda$, analyzed on two Poincaré surfaces. The main contributions show that at $\lambda=1.0$ the system exhibits highly symmetric OFM patterns, while for smaller $\lambda$ channel imbalance, figure-eight transport loops, and fractal-like transport boundaries emerge; LD ridges align with OFM boundaries, revealing the invariant-manifold skeleton driving transport. The results demonstrate that OFM and LD offer complementary, high-resolution tools for mapping transport in Hamiltonian systems and motivate applying the approach to other multi-channel potentials and time-dependent 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.

Figures (13)

• Figure 1: The unstretched ($\lambda = 1$) caldera potential energy surface $V(x,y)$ as defined in Eq. (\ref{['eq:caldera_potential']}), showing the central well and four saddle‐point exit channels located in each quadrant.
• Figure 2: Relative energy error over time using the ABA864 integrator for the time evolution of the orbit governed by the Hénon–Heiles Hamiltonian, Eq. \ref{['eq:hamiltonian']}, with initial conditions $\left(x_0,\, y_0,\, p_{y,0}\right) = \left(0,\, 0.1,\, 0.1\right)$, and $p_{x,0}$ determined from the energy constraint $\mathcal{H}(x_0, y_0, p_{x,0}, p_{y,0}) = \tfrac{1}{8}$. The integration time step is $\Delta t = 0.079$.
• Figure 3: Poincaré section of the system \ref{['eq:henon_heiles']} for $x = 0$ and $p_x > 0$ computed using the ABA864 integrator at fixed energy $H = 1/8$.
• Figure 4: Lagrangian Descriptor computation on the Poincaré section ($x=0$, $p_x>0$, $H=1/8$) for the HH system \ref{['eq:henon_heiles']}. Integration uses the ABA864 scheme with integration time step $dt=0.05$, $\tau=20$, grid $800\times800$ over $(y,p_y)\in[-0.6,0.6]$. (a) LD field; (b) 1D LD profile along $y=0$ indicated by a dashed vertical line in (a); (c) gradient magnitude $|\nabla \mathrm{LD}|$; (d) $|\nabla \mathrm{LD}|$ profile along $y=0$ indicated by a vertical dashed line in (c). The dashed line indicates the threshold $|\nabla \mathrm{LD}|=50$ used to identify manifold structures.
• Figure 5: Manifold set extracted from regions where $|\nabla \mathrm{LD}| > 50$, in Fig. \ref{['fig:ld_poincare_top']}. Thin ridges denote invariant manifolds and separatrices that bind transport regions.