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Bifurcation from a normally degenerate manifold
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Chillingworth, D. R. J. and Sbano, L. (2010) Bifurcation from a normally degenerate manifold. Proceedings of the London Mathematical Society, Vol.101 (No.1). pp. 137-178. doi:10.1112/plms/pdp055 ISSN 0024-6115 .
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Official URL: http://dx.doi.org/10.1112/plms/pdp055
Abstract
Local bifurcation theory typically deals with the response of a degenerate but isolated equilibrium state or periodic orbit of a dynamical system to perturbations controlled by one or more independent parameters, and characteristically uses tools from singularity theory. There are many situations, however, in which the equilibrium state or periodic orbit is not isolated but belongs to a manifold S of such states, typically as a result of continuous symmetries in the problem. In this case the bifurcation analysis requires a combination of local and global methods, and is most tractable in the case of normal nondegeneracy, that is, when the degeneracy is only along S itself and the system is nondegenerate in directions normal to S. In this paper we consider the consequences of relaxing normal nondegeneracy, which can generically occur within 1-parameter families of such systems. We pay particular attention to the simplest but important case where dim S=1 and where the normal degeneracy occurs with corank 1. Our main focus is on uniform degeneracy along S, although we also consider aspects of the branching structure for solutions when the degeneracy varies at different places on S. The tools are those of singularity theory adapted to global topology of S, which allow us to explain the bifurcation geometry in a natural way. In particular, we extend and give a clear geometric setting for earlier analytical results of Hale and Taboas.
Item Type: | Journal Article | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
Journal or Publication Title: | Proceedings of the London Mathematical Society | ||||
Publisher: | Cambridge University Press | ||||
ISSN: | 0024-6115 | ||||
Official Date: | July 2010 | ||||
Dates: |
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Volume: | Vol.101 | ||||
Number: | No.1 | ||||
Number of Pages: | 42 | ||||
Page Range: | pp. 137-178 | ||||
DOI: | 10.1112/plms/pdp055 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Restricted or Subscription Access | ||||
Funder: | London Mathematical Society | ||||
Grant number: | 4433 |
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