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The Classification of Bifurcations with Hidden Symmetries
Author(s) -
Manoel Míriam,
Stewart Ian
Publication year - 2000
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611500012156
Subject(s) - mathematics , equivariant map , neumann boundary condition , mathematical analysis , homogeneous space , singularity , bifurcation theory , tangent , singularity theory , pure mathematics , bifurcation , boundary value problem , nonlinear system , geometry , physics , quantum mechanics
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.