Stability and chaos in Kustaanheimo-Stiefel space induced by the Hopf fibration

TL;DR

This paper explores the topological and geometric structure of Kustaanheimo-Stiefel space via the Hopf fibration, analyzing stability and chaos in four-dimensional space and their implications for numerical simulations in celestial mechanics.

Contribution

It introduces a topological stability concept and a method to estimate the reliable simulation time scale in KS space, linking chaos and stability in a novel geometric framework.

Findings

Topological structure influences stability and chaos in KS space.

Numerical errors can break the topological structure, causing apparent chaos.

A method to estimate the trustworthy simulation time based on fiber separation.

Abstract

The need for the extra dimension in Kustaanheimo-Stiefel (KS) regularization is explained by the topology of the Hopf fibration, which defines the geometry and structure of KS space. A trajectory in Cartesian space is represented by a four-dimensional manifold, called the fundamental manifold. Based on geometric and topological aspects classical concepts of stability are translated to KS language. The separation between manifolds of solutions generalizes the concept of Lyapunov stability. The dimension-raising nature of the fibration transforms fixed points, limit cycles, attractive sets, and Poincar\'e sections to higher-dimensional subspaces. From these concepts chaotic systems are studied. In strongly perturbed problems the numerical error can break the topological structure of KS space: points in a fiber are no longer transformed to the same point in Cartesian space. An observer in three dimensions will see orbits departing from the same initial conditions but diverging in time. This apparent randomness of the integration can only be understood in four dimensions. The concept of topological stability results in a simple method for estimating the time scale in which numerical simulations can be trusted. Ideally all trajectories departing from the same fiber should be KS transformed to a unique trajectory in three-dimensional space, because the fundamental manifold that they constitute is unique. By monitoring how trajectories departing from one fiber separate from the fundamental manifold a critical time, equivalent to the Lyapunov time, is estimated. These concepts are tested on N-body examples: the Pythagorean problem, and an example of field stars interacting with a binary.