fold catastrophe

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1: 36.2 Catastrophes and Canonical Integrals

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Normal Forms Associated with Canonical Integrals: Cuspoid Catastrophe with Codimension $K$

… ►Special cases: $K=1$, fold catastrophe; $K=2$, cusp catastrophe; $K=3$, swallowtail catastrophe. …

2: 36.7 Zeros

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§36.7(i) Fold Canonical Integral

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3: 36.4 Bifurcation Sets

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Bifurcation (Catastrophe) Set for Cuspoids

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Bifurcation (Catastrophe) Set for Umbilics

… ► $K=1$, fold bifurcation set: … ►

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Figure 36.4.1: Bifurcation set of cusp catastrophe. Magnify

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Figure 36.4.2: Bifurcation set of swallowtail catastrophe. Magnify

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4: 36.10 Differential Equations

… ► $K=1$, fold: (36.10.1) becomes Airy’s equation (§9.2(i)) ►

36.10.3 $$\frac{{\partial }^2{\mathrm{\Psi }}_1}{{\partial x}^2}-\frac{x}{3}{\mathrm{\Psi }}_1=0.$$

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Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.10.E3

Encodings:

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See also:

Annotations for §36.10(i), §36.10(i), §36.10 and Ch.36

… ► $K=1$, fold: (36.10.6) is an identity. … ►

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5: 36.6 Scaling Relations

§36.6 Scaling Relations

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Diffraction Catastrophe Scaling

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${\mathrm{\Psi }}_K(𝐱;k)=k^{{\beta }_K}{\mathrm{\Psi }}_K\left(𝐲(k)\right),$

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${\mathrm{\Psi }}^{(\mathrm{U})}(𝐱;k)=k^{{\beta }^{(\mathrm{U})}}{\mathrm{\Psi }}^{(\mathrm{U})}\left({𝐲}^{(\mathrm{U})}(k)\right),$

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Table 36.6.1: Special cases of scaling exponents for cuspoids.

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singularity	$K$	${\beta }_K$	${\gamma }_{1K}$	${\gamma }_{2K}$	${\gamma }_{3K}$	${\gamma }_K$
fold	1	$\frac{1}{6}$	$\frac{2}{3}$	$-$	$-$	$\frac{2}{3}$
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