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:

hello

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.



Catastrophe sets of five elementary bifurcations.