Understanding the topological structure of phase space for dynamical systems in higher dimensions is critical for numerous applications, including the computation of chemical reaction rates and transport of objects in the solar system. Many topological techniques have been developed to study maps of two-dimensional (2D) phase spaces, but extending these techniques to higher dimensions is often a major challenge or even impossible. Previously, one such technique, homotopic lobe dynamics (HLD), was generalized to analyze the stable and unstable manifolds of hyperbolic fixed points for volume-preserving maps in three dimensions. This prior work assumed the existence of an equatorial heteroclinic intersection curve, which was the natural generalization of the 2D case. The present work extends the previous analysis to the case where no such equatorial curve exists, but where intersection curves, connecting fixed points may exist. In order to extend HLD to this case, we shift our perspective from the invariant manifolds of the fixed points to the invariant manifolds of the invariant circle formed by the fixed-point-to-fixed-point intersections. The output of the HLD technique is a symbolic description of the minimal underlying topology of the invariant manifolds. We demonstrate this approach through a series of examples.

了解更高维动力系统的相空间拓扑结构对于许多应用至关重要，包括计算化学反应速率和太阳系中物体的传输。已经开发了许多拓扑技术来研究二维 (2D) 相空间的映射，但将这些技术扩展到更高维度通常是一个重大挑战，甚至是不可能的。以前，一种这样的技术，同伦叶动力学（HLD），被推广到分析三维体积保持映射的双曲不动点的稳定和不稳定流形。这项先前的工作假设存在赤道异宿相交曲线，这是 2D 情况的自然概括。目前的工作将之前的分析扩展到不存在这样的赤道曲线，但可能存在相交曲线，连接不动点的情况。为了将 HLD 扩展到这种情况，我们将视角从不动点的不变流形转移到由不动点到不动点相交形成的不变圆的不变流形。HLD 技术的输出是不变流形的最小基础拓扑的符号描述。我们通过一系列示例演示了这种方法。HLD 技术的输出是不变流形的最小基础拓扑的符号描述。我们通过一系列示例演示了这种方法。HLD 技术的输出是不变流形的最小基础拓扑的符号描述。我们通过一系列示例演示了这种方法。