# swallowtail bifurcation set

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## 4 matching pages

##### 1: 36.4 Bifurcation Sets
$K=3$, swallowtail bifurcation set: …
###### §36.4(ii) Visualizations
36.11.4 $\Psi_{3}\left(x,0,0\right)=\frac{\sqrt{2\pi}}{(5|x|^{3})^{1/8}}\begin{cases}% \exp\left(-2\sqrt{2}(\ifrac{x}{5})^{5/4}\right)\left(\cos\left(2\sqrt{2}(% \ifrac{x}{5})^{5/4}-\tfrac{1}{8}\pi\right)+o\left(1\right)\right),&x\to+\infty% ,\\ \cos\left(4(\ifrac{|x|}{5})^{5/4}-\tfrac{1}{4}\pi\right)+o\left(1\right),&x\to% -\infty.\end{cases}$
36.11.5 $\Psi_{3}\left(0,y,0\right)=\overline{\Psi_{3}(0,-y,0)}=\exp\left(\tfrac{1}{4}i% \pi\right)\sqrt{\ifrac{\pi}{y}}\left(1-(i/{\sqrt{3}})\exp\left(\tfrac{3}{2}i(% \ifrac{2y}{5})^{5/3}\right)+o\left(1\right)\right),$ $y\to+\infty$.
36.11.6 $\Psi_{3}\left(0,0,z\right)=\frac{\Gamma\left(\tfrac{1}{3}\right)}{|z|^{1/3}% \sqrt{3}}+\begin{cases}o\left(1\right),&z\to+\infty,\\ \dfrac{2\sqrt{\pi}5^{1/4}}{(3|z|)^{3/4}}\left(\cos\left(\dfrac{2}{3}\left(% \dfrac{3|z|}{5}\right)^{5/2}-\dfrac{1}{4}\pi\right)+o\left(1\right)\right),&z% \to-\infty.\end{cases}$
Deep inside the bifurcation set, that is, inside the three-cusped astroid (36.4.10) and close to the part of the $z$-axis that is far from the origin, the zero contours form an array of rings close to the planes …The rings are almost circular (radii close to $(\Delta x)/9$ and varying by less than 1%), and almost flat (deviating from the planes $z_{n}$ by at most $(\Delta z)/36$). …Outside the bifurcation set (36.4.10), each rib is flanked by a series of zero lines in the form of curly “antelope horns” related to the “outside” zeros (36.7.2) of the cusp canonical integral. …
The zeros of these functions are curves in $\mathbf{x}=(x,y,z)$ space; see Nye (2007) for $\Phi_{3}$ and Nye (2006) for $\Phi^{(\mathrm{H})}$.