부산대학교 도서관

Author summary: ``Quantitative applications of elementary\break catastrophe theory require exact determination of the equivalence transformations involved. Let $\varphi(s;c)$ be an unfolding (which need not be universal) in $c\in{\bf R}^k$ of a cuspoid singularity $A_k$ in $s\in{\bf R}$. We 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 $\varphi$ only to degrees $l$ in $c$ and $j$ in $s$, satisfying $(l-m-1)k+j+1\leq0$. 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.''