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THE CLASSIFICATION OF BIFURCATIONS WITH HIDDEN SYMMETRIES

Published online by Cambridge University Press:  01 January 2000

MÍRIAM MANOEL  and
IAN STEWART
MÍRIAM MANOEL
Affiliation:
ICMC-USP, Caixa Postal 668, São Carlos, 13.560-970, [email protected]
IAN STEWART
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, [email protected]
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Abstract

We set up a singularity-theoretic framework for classifying one-parameter steady-state bifurcations with hidden symmetries. This framework also permits a non-trivial linearization at the bifurcation point. Many problems can be reduced to this situation; for instance, the bifurcation of steady or periodic solutions to certain elliptic partial differential equations with Neumann or Dirichlet boundary conditions. We formulate an appropriate equivalence relation with its associated tangent spaces, so that the usual methods of singularity theory become applicable. We also present an alternative method for computing those matrix-valued germs that appear in the equivalence relations employed in the classification of equivariant bifurcation problems. This result is motivated by hidden symmetries appearing in a class of partial differential equations defined on an $N$-dimensional rectangle under Neumann boundary conditions.

1991 Mathematics Subject Classification: 58C27, 58F14.

Keywords

hidden symmetrysingularitiesbifurcation
Type
Research Article
Information
Proceedings of the London Mathematical Society , Volume 80 , Issue 1 , January 2000 , pp. 198 - 234
Copyright
2000 London Mathematical Society

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