From The illustrated dictionary of nonlinear dynamics and chaos, T. Kapitaniak and S. R. Bishop, Wiley, 1999
Most smooth infinitely differentiable functions $V(x,c)$, where $x \in R^n, c \in R^k$, and $k\leq5$, are structurally stable. For this family $V:R^n \times R^k \to R$ and any point $(x,c) \in R^n \times R^k$ there is a choice of coordinates for $c$ in $R^k$ and for $x \in R^n$, such that $x$ varies smoothly with $c$, in terms which of the function $V(x,c)$ has one of the following local forms:
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a constant plus $$ x_1 $$ but not a fixed point $$ x^2_1+x^2_2+\ldots+x^2_r-\ldots+x^2_{r+1}-\ldots-x^2_n $$ or non-degenerate fixed point; Morse function $$ x^3_1+c_1x_1+(M) $$ fold catastrophe set $$ \pm(x^4_1+c_2x^2_1+c_1x_1)+(M) $$ cusp$(+)$, or dual cusp$(-)$ $$ x^5_1+\sum_{k=1}^{3}c_kx^k+(M) $$ swallowtail, $$ \pm(x^6_1+\sum_{k=1}^{4}c_kx^k)+(M) $$ butterfly or dual butterfly $$ x^7_1+\sum_{k=1}^{5}c_kx^k+(M) $$ wigman $$ x^2_1x_2\pm x^3_2+c_3x^2_1+c_2x_2+c_1x_1+(N) $$ hyperbolic $(+)$, or elliptic $(-)$ umbilic $$ \pm(x^2_1x_2+x^4_2+c_4x^2_2+c_3x^2_1+c_2x_2+c_1x_1)+(N) $$ parabolic (and dual) umbilic $$ \pm(x^2_1x_2+x^5_2+c_5x^3_2+c_4x^2_2+c_3x^2_1+c_2x_2+c_1x_1)+(N) $$ second hyperbolic $(+)$ and second elliptic $(-)$ umbilical $$ \pm(x^3_1+x^4_2+c_5x_1x^2_2+c_4x^2_2+c_3x_1x_2+c_2x_2+c_1x_1) $$ and symbolic (dual) umbilic.
(M) and (N) are given by
$$
(M)=x^2_2+\ldots+x^2_r-x^2_{r+1}-\dots-x^2_n
$$
and
$$
(N)=x^2_3+\ldots+x^2_r-x^2_{r+1}-\dots-x^2_n
$$
Remark: Thom’s theorem gives eleven elementary catastrophe sets (not counting duals). For k>5, the number of forms is infinite.
