Abstract
A Hopf bundle framework is constructed within Cn, in terms of which general paths on Cn ____ {0} are viewed and analyzed. The resulting hierarchy of spaces is addressed both theoretically and numerically, and the consequences for numerics and applications are investigated through a wide range of numerical experiments. The geometric reframing of Cn in this way - in terms of an intrinsic fibre bundle - allows for the introduction of bundle-theoretic quantities in a general dynamical setting. The roles of the various structural elements of the bundle are explored, including horizontal and vertical subspaces, parallel translation and connections. These concepts lead naturally to the association of a unique geometric phase with each path on Cn ____ {0}. This phase quantity is interpreted as a measure of the spinning in the S1 fibre of the Hopf bundle induced by paths on Cn ____ {0}, relative to a given connection, and is shown to be an important quantity. The implications of adopting this bundle viewpoint are investigated in two specific contexts. The first is the case of the lowest-dimensional Hopf bundle, S1 → S3 → S2. Here the quaternionic matrices are used to develop a simplified, geometrically intuitive formulation of the bundle structure, and a reduced expression for the phase is used to compute numerical phase results in three example systems. The second is the case where paths in Cn ____ {0} are generated by solutions to a particular class of parameter-dependent first-order ODEs. This establishes a direct link between the dynamical characteristics of such systems and the underlying bundle geometry. A variety of systems are examined and numerical phase results compiled. The numerics reveal an important correlation between the spectral properties of the path-generating ODEs and the resultant geometric phase change values. The details of this observed link are recorded in a conjecture.