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The need for the extra dimension in Kustaanheimo-Stiefel (KS) regularizationis explained by the topology of the Hopf fibration, which defines the geometryand structure of KS space. A trajectory in Cartesian space is represented by afour-dimensional manifold, called the fundamental manifold. Based on geometricand topological aspects classical concepts of stability are translated to KSlanguage. The separation between manifolds of solutions generalizes the conceptof Lyapunov stability. The dimension-raising nature of the fibration transformsfixed points, limit cycles, attractive sets, and Poincar\'e sections tohigher-dimensional subspaces. From these concepts chaotic systems are studied.In strongly perturbed problems the numerical error can break the topologicalstructure of KS space: points in a fiber are no longer transformed to the samepoint in Cartesian space. An observer in three dimensions will see orbitsdeparting from the same initial conditions but diverging in time. This apparentrandomness of the integration can only be understood in four dimensions. Theconcept of topological stability results in a simple method for estimating thetime scale in which numerical simulations can be trusted. Ideally alltrajectories departing from the same fiber should be KS transformed to a uniquetrajectory in three-dimensional space, because the fundamental manifold thatthey constitute is unique. By monitoring how trajectories departing from onefiber separate from the fundamental manifold a critical time, equivalent to theLyapunov time, is estimated. These concepts are tested on N-body examples: thePythagorean problem, and an example of field stars interacting with a binary.

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