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Title: Bifurcation problems with octahedral symmetry
Author: Melbourne, Ian
ISNI:       0000 0001 3392 7978
Awarding Body: University of Warwick
Current Institution: University of Warwick
Date of Award: 1987
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We analyse local bifurcation problems with octahedral symmetry using results from singularity theory. The thesis is split up into three sections. §1 comprises the bifurcation theory, and §3 contains a full singularity theory classification up to topological codimension one. The classification relies heavily upon new results about the recognition problem. These results are presented in §2 together with several examples drawn from equivariant bifurcation theory. These examples illustrate the new methods more clearly than the work in §3. In §1 we look at nondegenerate bifurcation problems equivariant with respect to the standard action of the octahedral group on Rɜ. We find three branches of symmetry-breaking bifurcation corresponding to the three maximal isotropy subgroups of the symmetry group with one-dimensional fixed-point subspaces. Locally, one of these branches is never stable, but precisely one of the other branches is stable if and only if all three branches bifurcate supercritically. In §2 we simplify the recognition problem by decomposing the group of equivalences into a unipotent group and a group of matrices. Building upon results of Bruce, du Plessis & Wall, we show that in many cases the unipotent problem can be solved by just using linear algebra. We give a necessary and sufficient condition for this, namely that the tangent space be invariant under unipotent equivalence. In addition we develop methods for checking whether the tangent space is invariant. The classification theorem in §3 gives a list of seven normal forms together with recognition problem solutions and universal unfoldings. Certain anomalies arise when comparing these results with those in Si. We reconcile the anomalies by giving a qualitative classification in addition to the standard classification. An application to barium titanate crystals is considered briefly.
Supervisor: Not available Sponsor: Science and Engineering Research Council
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA Mathematics