The University of Southampton
University of Southampton Institutional Repository

Bifurcations, singularities and symmetries

Bifurcations, singularities and symmetries
Bifurcations, singularities and symmetries
Bifurcation problems are typically about finding solutions x to an equation F(x,\mu)=0 as mu varies. In geometric terms we consider a smooth map F:N\times K\to Q$ between manifolds (thinking of elements of N as it variables and those of K as it parameters, choose a particular point q\in Q (which we may take to be 0\i\bf R^n$) and ask how the solution set M_\mu=F_\mu^-1(0)$ varies as \mu varies in K, with the notation that F_\mu(x)=F(x,\mu). The structure of this problem is captured by the geometry of the projection map pi:M\to K where M=F^-1(0) and where \pi is the projection into the parameter space K. In particular, the bifurcation set (the locus in K across which the configuration of solutions x in N may change qualitatively) is the set of singular values of pi. We discuss generic aspects of this geometry, with special emphasis on cases where discrete and/or continuous (it e.g. rotational) symmetries are involved.
85-92
Chillingworth, David
39d011b7-db33-4d7d-8dc7-c5a4e0a61231
Chillingworth, David
39d011b7-db33-4d7d-8dc7-c5a4e0a61231

Chillingworth, David (2000) Bifurcations, singularities and symmetries. Summer School on Differential Geometry, Portugal. 03 - 07 Sep 1999. pp. 85-92 .

Record type: Conference or Workshop Item (Other)

Abstract

Bifurcation problems are typically about finding solutions x to an equation F(x,\mu)=0 as mu varies. In geometric terms we consider a smooth map F:N\times K\to Q$ between manifolds (thinking of elements of N as it variables and those of K as it parameters, choose a particular point q\in Q (which we may take to be 0\i\bf R^n$) and ask how the solution set M_\mu=F_\mu^-1(0)$ varies as \mu varies in K, with the notation that F_\mu(x)=F(x,\mu). The structure of this problem is captured by the geometry of the projection map pi:M\to K where M=F^-1(0) and where \pi is the projection into the parameter space K. In particular, the bifurcation set (the locus in K across which the configuration of solutions x in N may change qualitatively) is the set of singular values of pi. We discuss generic aspects of this geometry, with special emphasis on cases where discrete and/or continuous (it e.g. rotational) symmetries are involved.

PDF
pdgchil.pdf - Accepted Manuscript
Download (6MB)

More information

Published date: 2000
Venue - Dates: Summer School on Differential Geometry, Portugal, 1999-09-03 - 1999-09-07

Identifiers

Local EPrints ID: 29791
URI: http://eprints.soton.ac.uk/id/eprint/29791
PURE UUID: dacf33b8-b2fe-4455-a055-1250eb31c41b

Catalogue record

Date deposited: 16 Mar 2007
Last modified: 17 Jul 2017 15:56

Export record

Download statistics

Downloads from ePrints over the past year. Other digital versions may also be available to download e.g. from the publisher's website.

View more statistics

Atom RSS 1.0 RSS 2.0

Contact ePrints Soton: eprints@soton.ac.uk

ePrints Soton supports OAI 2.0 with a base URL of http://eprints.soton.ac.uk/cgi/oai2

This repository has been built using EPrints software, developed at the University of Southampton, but available to everyone to use.

We use cookies to ensure that we give you the best experience on our website. If you continue without changing your settings, we will assume that you are happy to receive cookies on the University of Southampton website.

×