Dynamics and numerics on the Hopf bundle

 Department of Mathematics
 University of Surrey, UK

Thomas J. Bridges & Rupert Way

Dynamics in the Hopf bundle and the geometric phase
  A Hopf bundle framework is constructed within C^n, in terms of which general paths on C^n\{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 C^n 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 C^n\{0}. This phase quantity is interpreted as a measure of the spinning in the S^1 fibre of the Hopf bundle induced by paths on C^n\{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, S^1 --> S^3 --> S^2. 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 C^n\{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.
  R. Way. Dynamics in the Hopf bundle, the geometric phase and implications for dynamical systems. PhD Thesis, University of Surrey (2009)  

Proof of the connection between geometric phase and wave stability
  A stability index is developed for the traveling waves of non-linear reaction-diffusion equations using the geometric phase induced on the Hopf bundle. This can be viewed as an alternative formulation of the winding number calculation of the Evans function, whose zeros correspond to the eigenvalues of the linearization of reaction-diffusion operators about the wave. The stability of a traveling wave can be determined by the existence of eigenvalues of positive real part for the linear operator. Our method of geometric phase for locating and counting eigenvalues is inspired by the numerical results in Way’s (2009) thesis on dynamics in the Hopf bundle. A detailed proof is provided of the relationship between the phase and eigenvalues for dynamical systems defined on 2D and sketch the proof of the method of geometric phase for N-dimensions and its generalization to boundary-value problems. Implementing the numerical method, modified from Way (2009), open questions are flagged up, inspired by the results.
  G.J. Grudzien, T.J. Bridges, & C.K.R.T. Jones. Geometric phase in the Hopf bundle and the stability of nonlinear waves Physica D (in press, 2016). In press, corrected proof

Introduction to the theory of connections
  The basic facts of bundle theory are recorded. Several pictures are presented, which should be thought of as analogies for the actual mathematical structures involved in the theory. The motivation for including quite so many illustrations here is that, in general, diagrams are conspicuously absent from texts on the subject, yet are central to an intuitive, geometric understanding of the subject. This novel angle is important because many of the structures involved in bundles can be described in different but equivalent ways mathematically (dual representations), and this duality is often a lot easier to understand in the context of pictures of well-known geometrical structures.
  R. Way. Introduction to connections on principal fibre bundles. Report, University of Surrey  

Parallel transport on the two-sphere
  One of the simplest examples of parallel transport is that of a vector around a loop on the surface of a 2-sphere. Consider the case where a vector is tangent to the sphere at some point on the equator (marked A on the figure to the right) and also tangent to a given great circle between the north and south poles. Have the tangent vector move towards the north pole as shown. Now parallel transport it along another great circle down to the equator (point B), and then along the equator back to point A. Upon completion of its circuit on the sphere, parameterized by coordinates of longitude and latitude, the vector fails to return to its original state. It is pointing in a different direction. This difference is called holonomy (by mathematicians and anholonomy by the physicists). The difference in angle between the initial and the final vectors at the north pole is proportional to the solid angle subtended by the circuit on the sphere and is independent of the particular coordinatization. It is a feature of the geometry of the surface of the sphere.
  R.W. Batterman. Falling cats, parallel parking, and polarized light. Studies in History and Philosophy of Science, Part B: Studies In History and Philosophy of Modern Physics 34 527-557 (2003). 

Matlab codes for integrating on the Hopf bundle
Complex orthogonalization and Stiefel bundles
  Numerical integration of complex linear systems of ODEs depending analytically on an eigenvalue parameter are considered. Complex orthogonalization, which is required to stabilize the numerical integration, results in non-analytic systems. It is shown that properties of eigenvalues are still efficiently recoverable by extracting information from a non-analytic characteristic function. The orthonormal systems are constructed using the geometry of Stiefel bundles. Different forms of continuous orthogonalization in the literature are shown to correspond to different choices of connection one-form on the Stiefel bundle. For the numerical integration, Gauss-Legendre Runge-Kutta algorithms are the principal choice for preserving orthogonality, and performance results are shown for a range of GLRK methods. The theory and methods are tested by application to example boundary value problems including the Orr-Sommerfeld equation in hydrodynamic stability.
  D. Avitabile & T.J. Bridges. Numerical implementation of complex orthogonalization, parallel transport on Stiefel bundles, and analyticity Physica D 239 1038-1047 (2010). 
Surrey Preprint .pdf

The Chern number and holomorphic boundary value problems
  Holomorphic families of linear ordinary differential equations on a finite interval with prescribed parameter-dependent boundary conditions are considered from a geometrical viewpoint. The Gardner-Jones bundle, which was introduced for linearized reaction-diffusion equations, is generalized and applied to this abstract class of lambda-dependent boundary-value problems (BVPs), where lambda is a complex eigenvalue parameter. The fundamental analytical object of such BVPs is the characteristic determinant, and it is proved that any characteristic determinant on a Jordan curve can be characterized geometrically as the determinant of a transition function associated with the Gardner-Jones bundle. The topology of the bundle, represented by the Chern number, then yields precise information about the number of eigenvalues in a prescribed subset of the complex lambda-plane. This result shows that the Gardner-Jones bundle is an intrinsic geometric property of such lambda-dependent BVPs. The bundle framework is applied to examples from hydrodynamic stability theory and the linearized complex Ginzburg-Landau equation.
  F. R. Austin & T. J. Bridges. A bundle view of boundary-value problems: generalizing the Gardner-Jones bundle J. Diff. Eqns 189 412-439 (2003). 

  Department of Mathematics     University of Surrey