UC Merced

Understanding the topological structure of phase space for dynamical systems inhigher dimensions is critical for numerous applications, including transport of objects in the solar system, systems of uids, and charged particles in crossed magnetic and electric elds. 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. One such technique, homotopic lobe dynamics (HLD), has shown great success in analyzing the stable and unstable manifolds of hyperbolic xed points for area-preserving maps in two dimensions. The output of the HLD technique is a symbolic description of the minimal underlying topology of the invariant manifolds. The present work extends HLD to volume-preserving maps in three dimensions. We extend HLD to systems that have equatorial heteroclinic intersections, pole-to-pole invariant circles, and forced pole-to-pole heteroclinic intersections. In order to extend HLD to these cases, we went through multiple computational methodologies as well shift our perspective of manifolds in 3D. We demonstrate the power of the HLD by applying it to increasingly complex numerical and theoretical examples.