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.

我们为三自由度时间无关的哈密顿系统开发了反应岛理论的几何、分析和计算框架。在这种情况下，动力学发生在相空间的 5 维能量表面中，并由 3 维常双曲不变球的 4 维稳定和不稳定流形控制。稳定和不稳定流形具有球体的几何结构，我们提供了研究这些球体及其相交点决定轨迹动力学演化方式的方法。该几何图是通过拉格朗日描述符的计算技术实现的。在一组轨迹中，拉格朗日描述符允许我们识别最接近稳定或不稳定流形的那些。