Bifurcations with spherical symmetry

• Main Author: Sigrist, Rachel
• Published: University of Nottingham 2010
• Subjects: 510; QA299 Analysis
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.523612

Bifurcations from spherically symmetric states can occur in many physical and biological systems. These include the development of a spherical ball of cells into an asymmetrical state and the buckling of a sphere under pressure. They also occur in the evolution of reaction–diffusion systems on a spherical surface and in Rayleigh–Benard convection in a spherical shell. Many of the behaviours of these systems can be explained by their underlying spherical symmetry alone. Using results from the area of mathematics known as equivariant bifurcation theory we can use group theoretical methods both to predict the symmetries of the solutions which are expected to result from bifurcations with symmetry and compute their stability. In this thesis both stationary and Hopf bifurcation with spherical symmetry are discussed. Firstly, using group theoretical techniques, the symmetries of the periodic solutions which can be found at a Hopf bifurcation with spherical symmetry are computed. This computation has been carried out previously but contains some errors which are corrected here. For one particular representation of the group of symmetries of the sphere the stability properties of the bifurcating branches of periodic solutions resulting from the Hopf bifurcation are analysed and a survey is carried out of other periodic and quasiperiodic solutions which can exist. Secondly, symmetry considerations are used to investigate the existence and stability properties of symmetric spiral patterns on the surface of a sphere which result from stationary bifurcations. It is found that in the case of the Swift–Hohenberg equation spiral patterns with one or more arms can exist and be stable on spheres of certain radii. Although one-armed spirals in the Swift–Hohenberg equation are stationary solutions, it is shown that generically one-armed spirals on spheres must drift.