On the exact reduction of a univariate catastrophe to normal form

Abstract

Quantitative applications for elementary catastrophe theory require exact determination of the equivalence transformations involved. Let phi (s;c) be an unfolding (which need not be universal) in c in Rk of a cuspoid singularity Ak in s in R. The authors discuss its reduction via a sequence of coordinate transformations to normal form, exact to degree m in the control variables c, and show that this requires knowledge of the terms of phi only to degrees l in c and j in s satisfying (l-m-1)k+j+1<or=0. The 'linear normal form', which describes the orientation and shear of the bifurcation set, is discussed in detail, and normal form methods for deriving tangent spaces and curvatures of singularity manifolds are described, with examples.