Bifurcations with spherical symmetry

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 sph...

Full description

Bibliographic Details
Main Author: Sigrist, Rachel
Published: University of Nottingham 2010
Subjects:
510
Online Access:http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.523612
id ndltd-bl.uk-oai-ethos.bl.uk-523612
record_format oai_dc
spelling ndltd-bl.uk-oai-ethos.bl.uk-5236122015-03-20T03:18:13ZBifurcations with spherical symmetrySigrist, Rachel2010Bifurcations 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.510QA299 AnalysisUniversity of Nottinghamhttp://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.523612http://eprints.nottingham.ac.uk/11170/Electronic Thesis or Dissertation
collection NDLTD
sources NDLTD
topic 510
QA299 Analysis
spellingShingle 510
QA299 Analysis
Sigrist, Rachel
Bifurcations with spherical symmetry
description 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.
author Sigrist, Rachel
author_facet Sigrist, Rachel
author_sort Sigrist, Rachel
title Bifurcations with spherical symmetry
title_short Bifurcations with spherical symmetry
title_full Bifurcations with spherical symmetry
title_fullStr Bifurcations with spherical symmetry
title_full_unstemmed Bifurcations with spherical symmetry
title_sort bifurcations with spherical symmetry
publisher University of Nottingham
publishDate 2010
url http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.523612
work_keys_str_mv AT sigristrachel bifurcationswithsphericalsymmetry
_version_ 1716779677739122688