Reactive Islands for Three Degrees-of-Freedom Hamiltonian Systems

TL;DR

This paper introduces a comprehensive framework combining geometry, analysis, and computation to study reactive islands in three degrees-of-freedom Hamiltonian systems, enhancing understanding of complex dynamical behaviors.

Contribution

It develops a novel geometrical and computational approach for analyzing stable and unstable manifolds in 3-DoF Hamiltonian systems using Lagrangian descriptors.

Findings

Identifies stable and unstable manifolds using Lagrangian descriptors.

Provides a method to compute flux between regions on the energy surface.

Characterizes the geometry of spherinders and their intersections.

Abstract

We develop the geometrical, analytical, and computational framework for reactive island theory for three degrees-of-freedom time-independent Hamiltonian systems. In this setting, the dynamics occurs in a 5-dimensional energy surface in phase space and is governed by four-dimensional stable and unstable manifolds of a three-dimensional normally hyperbolic invariant sphere. The stable and unstable manifolds have the geometrical structure of spherinders and we provide the means to investigate the ways in which these spherinders and their intersections determine the dynamical evolution of trajectories. This geometrical picture is realized through the computational technique of Lagrangian descriptors. In a set of trajectories, Lagrangian descriptors allow us to identify the ones closest to a stable or unstable manifold. Using an approximation of the manifold on a surface of section we are able to calculate the flux between two regions of the energy surface.