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

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