Citation :: Thom's Theorem

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 $k\le 5$, 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 $$x_1^2+x_2^2+\dots +x_r^2-\dots +x_{r+1}^2-\dots -x_n^2$$ or non-degenerate fixed point; Morse function $$x_1^3+c_1{x}_1+(M)$$ fold catastrophe set $$\pm (x_1^4+c_2{x}_1^2+c_1{x}_1)+(M)$$ cusp$(+)$, or dual cusp$(-)$ $$x_1^5+\sum _{k=1}^3{c}_k{x}^k+(M)$$ swallowtail, $$\pm (x_1^6+\sum _{k=1}^4{c}_k{x}^k)+(M)$$ butterfly or dual butterfly $$x_1^7+\sum _{k=1}^5{c}_k{x}^k+(M)$$ wigman $$x_1^2{x}_2\pm x_2^3+c_3{x}_1^2+c_2{x}_2+c_1{x}_1+(N)$$ hyperbolic $(+)$, or elliptic $(-)$ umbilic $$\pm (x_1^2{x}_2+x_2^4+c_4{x}_2^2+c_3{x}_1^2+c_2{x}_2+c_1{x}_1)+(N)$$ parabolic (and dual) umbilic $$\pm (x_1^2{x}_2+x_2^5+c_5{x}_2^3+c_4{x}_2^2+c_3{x}_1^2+c_2{x}_2+c_1{x}_1)+(N)$$ second hyperbolic $(+)$ and second elliptic $(-)$ umbilical $$\pm (x_1^3+x_2^4+c_5{x}_1{x}_2^2+c_4{x}_2^2+c_3{x}_1{x}_2+c_2{x}_2+c_1{x}_1)$$ and symbolic (dual) umbilic.

(M) and (N) are given by $$(M)=x_2^2+\dots +x_r^2-x_{r+1}^2-\cdots -x_n^2$$ and $$(N)=x_3^2+\dots +x_r^2-x_{r+1}^2-\cdots -x_n^2$$

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.