# §36.4 Bifurcation Sets

## §36.4(i) Formulas

### Critical Points for Cuspoids

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

 36.4.1 $\frac{\partial}{\partial t}\mathop{\Phi_{K}\/}\nolimits\!\left(t_{j}(\mathbf{x% });\mathbf{x}\right)=0.$

### Critical Points for Umbilics

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 $\displaystyle\frac{\partial}{\partial s}\mathop{\Phi^{(\mathrm{U})}\/}% \nolimits\!\left(s_{j}(\mathbf{x}),t_{j}(\mathbf{x});\mathbf{x}\right)$ $\displaystyle=0,$ $\displaystyle\frac{\partial}{\partial t}\mathop{\Phi^{(\mathrm{U})}\/}% \nolimits\!\left(s_{j}(\mathbf{x}),t_{j}(\mathbf{x});\mathbf{x}\right)$ $\displaystyle=0.$

### Bifurcation (Catastrophe) Set for Cuspoids

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

 36.4.3 $\frac{{\partial}^{2}}{{\partial t}^{2}}\mathop{\Phi_{K}\/}\nolimits\!\left(t;% \mathbf{x}\right)=0.$

### Bifurcation (Catastrophe) Set for Umbilics

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

 36.4.4 $\frac{{\partial}^{2}}{{\partial s}^{2}}\mathop{\Phi^{(\mathrm{U})}\/}\nolimits% \!\left(s,t;\mathbf{x}\right)\frac{{\partial}^{2}}{{\partial t}^{2}}\mathop{% \Phi^{(\mathrm{U})}\/}\nolimits\!\left(s,t;\mathbf{x}\right)-\left(\frac{{% \partial}^{2}}{\partial s\partial t}\mathop{\Phi^{(\mathrm{U})}\/}\nolimits\!% \left(s,t;\mathbf{x}\right)\right)^{2}=0.$

### Special Cases

$K=1$, 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)

$K=2$, 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)

$K=3$, swallowtail bifurcation set:

 36.4.7 $\displaystyle x$ $\displaystyle=3t^{2}(z+5t^{2}),$ $\displaystyle y$ $\displaystyle=-t(3z+10t^{2})$, $-\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)

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)

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)

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

 36.4.10 $\displaystyle x$ $\displaystyle=\tfrac{1}{3}z^{2}(-\mathop{\cos\/}\nolimits\!\left(2\phi\right)-% 2\mathop{\cos\/}\nolimits\phi),$ $\displaystyle y$ $\displaystyle=\tfrac{1}{3}z^{2}(\mathop{\sin\/}\nolimits\!\left(2\phi\right)-2% \mathop{\sin\/}\nolimits\phi)$, $0\leq\phi\leq 2\pi$.

Elliptic umbilic cusp lines (ribs):

 36.4.11 $x+iy=-z^{2}\mathop{\exp\/}\nolimits\!\left(\tfrac{2}{3}i\pi m\right),$ $m=0,1,2$.

Hyperbolic umbilic bifurcation set (codimension three):

 36.4.12 $\displaystyle x$ $\displaystyle=-\tfrac{1}{12}z^{2}(\mathop{\exp\/}\nolimits\!\left(2\tau\right)% \pm 2\mathop{\exp\/}\nolimits\!\left(-\tau\right)),$ $\displaystyle y$ $\displaystyle=-\tfrac{1}{12}z^{2}(\mathop{\exp\/}\nolimits\!\left(-2\tau\right% )\pm 2\mathop{\exp\/}\nolimits\!\left(\tau\right))$, $-\infty\leq\tau<\infty.$ Symbols: $\mathop{\exp\/}\nolimits\NVar{z}$: 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)

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)

For derivations of the results in this subsection see Poston and Stewart (1978, Chapter 9).