cusp catastophe

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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. … ►

Canonical Integrals

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2: 36.5 Stokes Sets

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$K=2$. Cusp

►The Stokes set is itself a cusped curve, connected to the cusp of the bifurcation set: … ►They generate a pair of cusp-edged sheets connected to the cusped sheets of the swallowtail bifurcation set (§36.4). … ►

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Figure 36.5.1: Cusp catastrophe. Magnify

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3: 36.7 Zeros

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§36.7(ii) Cusp Canonical Integral

… ► … ►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 …Away from the $z$-axis and approaching the cusp lines (ribs) (36.4.11), the lattice becomes distorted and the rings are deformed, eventually joining to form “hairpins” whose arms become the pairs of zeros (36.7.1) of the cusp canonical integral. …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. …

4: 36.4 Bifurcation Sets

… ► $K=2$, cusp bifurcation set: … ►Swallowtail cusp lines (ribs): … ►Elliptic umbilic bifurcation set (codimension three): for fixed $z$, the section of the bifurcation set is a three-cusped astroid … ►Elliptic umbilic cusp lines (ribs): … ►Hyperbolic umbilic cusp line (rib): …

5: 23.15 Definitions

… ►If, in addition, $f(\tau )\to 0$ as $q\to 0$, then $f(\tau )$ is called a cusp form. …

6: 23.18 Modular Transformations

… ►and $\lambda \left(\tau \right)$ is a cusp form of level zero for the corresponding subgroup of SL$(2,\mathbb{Z})$. …

7: 36.6 Scaling Relations

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Table 36.6.1: Special cases of scaling exponents for cuspoids.

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singularity	$K$	${\beta }_K$	${\gamma }_{1K}$	${\gamma }_{2K}$	${\gamma }_{3K}$	${\gamma }_K$
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cusp	2	$\frac{1}{4}$	$\frac{3}{4}$	$\frac{1}{2}$	$-$	$\frac{5}{4}$
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8: Bibliography F

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Transverse cusp diffraction catastrophes produced by the reflection of ultrasonic tone bursts from a curved surface in water. J. Acoust. Soc. Amer. 92 (5), pp. 2869–2877.

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External Links:

Document

Referenced by:

§36.14(iv).

Encodings:

BibTeX

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Travel time surface of a transverse cusp caustic produced by reflection of acoustical transients from a curved metal surface. J. Acoust. Soc. Amer. 95 (2), pp. 650–660.

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External Links:

Document

Referenced by:

§36.14(iv).

Encodings:

BibTeX

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9: 36.10 Differential Equations

… ► $K=2$, cusp: … ► $K=2$, cusp: …

10: Bibliography P

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The structure of an electromagnetic field in the neighbourhood of a cusp of a caustic. Philos. Mag. (7) 37, pp. 311–317.

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External Links:

MathReview (C. J. Bouwkamp)

Referenced by:

§36.2(ii).

Encodings:

BibTeX

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