36 Integrals with Coalescing Saddles Properties 36.3 Visualizations of Canonical Integrals 36.5 Stokes Sets

§36.4 Bifurcation Sets

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Keywords:: bifurcation sets
Referenced by:: §36.5(ii), §36.5(ii)
Permalink:: http://dlmf.nist.gov/36.4
See also:: Annotations for Ch.36

§36.4(i) Formulas

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Keywords:: critical points
Permalink:: http://dlmf.nist.gov/36.4.i
See also:: Annotations for §36.4 and Ch.36

Critical Points for Cuspoids

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See also:: Annotations for §36.4(i), §36.4 and Ch.36

These are real solutions $t_{j}(\mathbf{x})$ , $1\leq j\leq j_{\max}(\mathbf{x})\leq K+1$ , of

36.4.1		$\frac{\partial}{\partial t}\Phi_{K}\left(t_{j}(\mathbf{x});\mathbf{x}\right)=0.$
ⓘ Symbols: $\Phi_{\NVar{K}}\left(\NVar{t};\NVar{\mathbf{x}}\right)$ : cuspoid catastrophe of codimension $K$ , $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $t$ : variable, $K$ : codimension and $t_{j}(\mathbf{x})$ : solutions Referenced by: §36.11, §36.12(i), §36.4(i), §36.5(i), §36.5(ii) Permalink: http://dlmf.nist.gov/36.4.E1 Encodings: TeX, pMML, png See also: Annotations for §36.4(i), §36.4(i), §36.4 and Ch.36

Critical Points for Umbilics

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See also:: Annotations for §36.4(i), §36.4 and Ch.36

These are real solutions $\{s_{j}(\mathbf{x}),t_{j}(\mathbf{x})\}$ , $1\leq j\leq j_{\max}(\mathbf{x})\leq 4$ , of

36.4.2	$\displaystyle\frac{\partial}{\partial s}\Phi^{(\mathrm{U})}\left(s_{j}(\mathbf% {x}),t_{j}(\mathbf{x});\mathbf{x}\right)$	$\displaystyle=0,$
36.4.2	$\displaystyle\frac{\partial}{\partial t}\Phi^{(\mathrm{U})}\left(s_{j}(\mathbf% {x}),t_{j}(\mathbf{x});\mathbf{x}\right)$	$\displaystyle=0.$
ⓘ Symbols: $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $\Phi^{(\mathrm{U})}\left(\NVar{s},\NVar{t};\NVar{\mathbf{x}}\right)$ : elliptic umbilic catastrophe for $\mathrm{U}=\mathrm{E}\mbox{ or }\mathrm{K}$ , $t$ : variable, $s$ : variable and $t_{j}(\mathbf{x})$ : solutions Referenced by: §36.4(i), §36.5(i) Permalink: http://dlmf.nist.gov/36.4.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §36.4(i), §36.4(i), §36.4 and Ch.36

Bifurcation (Catastrophe) Set for Cuspoids

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See also:: Annotations for §36.4(i), §36.4 and Ch.36

This is the codimension-one surface in $\mathbf{x}$ space where critical points coalesce, satisfying (36.4.1) and

36.4.3		$\frac{{\partial}^{2}}{{\partial t}^{2}}\Phi_{K}\left(t;\mathbf{x}\right)=0.$
ⓘ Symbols: $\Phi_{\NVar{K}}\left(\NVar{t};\NVar{\mathbf{x}}\right)$ : cuspoid catastrophe of codimension $K$ , $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $t$ : variable and $K$ : codimension Permalink: http://dlmf.nist.gov/36.4.E3 Encodings: TeX, pMML, png See also: Annotations for §36.4(i), §36.4(i), §36.4 and Ch.36

Bifurcation (Catastrophe) Set for Umbilics

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See also:: Annotations for §36.4(i), §36.4 and Ch.36

This is the codimension-one surface in $\mathbf{x}$ space where critical points coalesce, satisfying (36.4.2) and

36.4.4		$\frac{{\partial}^{2}}{{\partial s}^{2}}\Phi^{(\mathrm{U})}\left(s,t;\mathbf{x}% \right)\frac{{\partial}^{2}}{{\partial t}^{2}}\Phi^{(\mathrm{U})}\left(s,t;% \mathbf{x}\right)-\left(\frac{\,{\partial}^{2}}{\,\partial s\,\partial t}\Phi^% {(\mathrm{U})}\left(s,t;\mathbf{x}\right)\right)^{2}=0.$
ⓘ Symbols: $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $\Phi^{(\mathrm{U})}\left(\NVar{s},\NVar{t};\NVar{\mathbf{x}}\right)$ : elliptic umbilic catastrophe for $\mathrm{U}=\mathrm{E}\mbox{ or }\mathrm{K}$ , $t$ : variable and $s$ : variable Permalink: http://dlmf.nist.gov/36.4.E4 Encodings: TeX, pMML, png See also: Annotations for §36.4(i), §36.4(i), §36.4 and Ch.36

Special Cases

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Keywords:: bifurcation set, cusp bifurcation set, elliptic umbilic bifurcation set, fold canonical integral, formula, hyperbolic umbilic bifurcation set, swallowtail bifurcation set
See also:: Annotations for §36.4(i), §36.4 and Ch.36

$K=1$ , fold bifurcation set:

36.4.5		$x=0.$
ⓘ Symbols: $x$ : real parameter Permalink: http://dlmf.nist.gov/36.4.E5 Encodings: TeX, pMML, png See also: Annotations for §36.4(i), §36.4(i), §36.4 and Ch.36

$K=2$ , cusp bifurcation set:

36.4.6		$27x^{2}=-8y^{3}.$
ⓘ Symbols: $y$ : real parameter and $x$ : real parameter Permalink: http://dlmf.nist.gov/36.4.E6 Encodings: TeX, pMML, png See also: Annotations for §36.4(i), §36.4(i), §36.4 and Ch.36

$K=3$ , swallowtail bifurcation set:

36.4.7		$\displaystyle x$	$\displaystyle=3t^{2}(z+5t^{2}),$
		$\displaystyle y$	$\displaystyle=-t(3z+10t^{2})$ ,
	$-\infty<t<\infty$ .
ⓘ Symbols: $y$ : real parameter, $z$ : real parameter, $t$ : variable and $x$ : real parameter Permalink: http://dlmf.nist.gov/36.4.E7 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §36.4(i), §36.4(i), §36.4 and Ch.36

Swallowtail self-intersection line:

36.4.8	$\displaystyle y$	$\displaystyle=0,$
	$\displaystyle z$	$\displaystyle\leq 0,$
	$\displaystyle x$	$\displaystyle=\tfrac{9}{20}z^{2}.$
ⓘ Symbols: $y$ : real parameter, $z$ : real parameter and $x$ : real parameter Permalink: http://dlmf.nist.gov/36.4.E8 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §36.4(i), §36.4(i), §36.4 and Ch.36

Swallowtail cusp lines (ribs):

36.4.9	$\displaystyle z$	$\displaystyle\leq 0,$
	$\displaystyle x$	$\displaystyle=-\tfrac{3}{20}z^{2},$
	$\displaystyle 10y^{2}$	$\displaystyle=-4z^{3}.$
ⓘ Symbols: $y$ : real parameter, $z$ : real parameter and $x$ : real parameter Permalink: http://dlmf.nist.gov/36.4.E9 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §36.4(i), §36.4(i), §36.4 and Ch.36

Elliptic umbilic bifurcation set (codimension three): for fixed $z$ , the section of the bifurcation set is a three-cusped astroid

36.4.10		$\displaystyle x$	$\displaystyle=\tfrac{1}{3}z^{2}(-\cos\left(2\phi\right)-2\cos\phi),$
		$\displaystyle y$	$\displaystyle=\tfrac{1}{3}z^{2}(\sin\left(2\phi\right)-2\sin\phi)$ ,
	$0\leq\phi\leq 2\pi$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $y$ : real parameter, $z$ : real parameter and $x$ : real parameter Referenced by: §36.7(iii) Permalink: http://dlmf.nist.gov/36.4.E10 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §36.4(i), §36.4(i), §36.4 and Ch.36

Elliptic umbilic cusp lines (ribs):

36.4.11		$x+iy=-z^{2}\exp\left(\tfrac{2}{3}i\pi m\right),$
$m=0,1,2$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $y$ : real parameter, $z$ : real parameter, $n$ : integer and $x$ : real parameter Referenced by: §36.7(iii) Permalink: http://dlmf.nist.gov/36.4.E11 Encodings: TeX, pMML, png See also: Annotations for §36.4(i), §36.4(i), §36.4 and Ch.36

Hyperbolic umbilic bifurcation set (codimension three):

36.4.12		$\displaystyle x$	$\displaystyle=-\tfrac{1}{12}z^{2}(\exp\left(2\tau\right)\pm 2\exp\left(-\tau% \right)),$
		$\displaystyle y$	$\displaystyle=-\tfrac{1}{12}z^{2}(\exp\left(-2\tau\right)\pm 2\exp\left(\tau% \right))$ ,
	$-\infty\leq\tau<\infty.$
ⓘ Symbols: $\exp\NVar{z}$ : exponential function, $y$ : real parameter, $z$ : real parameter and $x$ : real parameter Permalink: http://dlmf.nist.gov/36.4.E12 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §36.4(i), §36.4(i), §36.4 and Ch.36

The $+$ sign labels the cusped sheet; the $-$ sign labels the sheet that is smooth for $z\not=0$ (see Figure 36.4.4).

Hyperbolic umbilic cusp line (rib):

36.4.13		$x=y=-\tfrac{1}{4}z^{2}.$
ⓘ Symbols: $y$ : real parameter, $z$ : real parameter and $x$ : real parameter Permalink: http://dlmf.nist.gov/36.4.E13 Encodings: TeX, pMML, png See also: Annotations for §36.4(i), §36.4(i), §36.4 and Ch.36