The geometry of generic k-parameter bifurcation form an n-manifold is discussed for all values of k, n with particular emphasis on the case n=2 (the case n=1 being dealt with in earlier work). Such bifurcations typically arise in the study of equilibrium states of dynamical systems with continuous (for example, spherical or toroidal) symmetry which undergo small symmetry-breaking perturbations, and in the use of Melinkov maps for detecting bifurcations of periodic orbits from resonance. Detailed analysis is given in the interesting case n=2, k=3 where the local geometry partly resembles unfolding of a degenerate wavefront or Legendrian collapse.
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