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1: 36.2 Catastrophes and Canonical Integrals
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
§36.7(i) Fold Canonical Integral
3: 36.4 Bifurcation Sets
Bifurcation (Catastrophe) Set for Cuspoids
Bifurcation (Catastrophe) Set for Umbilics
K = 1 , fold bifurcation set: …
See accompanying text
Figure 36.4.1: Bifurcation set of cusp catastrophe. Magnify
See accompanying text
Figure 36.4.2: Bifurcation set of swallowtail catastrophe. Magnify
4: 36.10 Differential Equations
K = 1 , fold: (36.10.1) becomes Airy’s equation (§9.2(i)) … K = 1 , fold: (36.10.6) is an identity. …
Φ s ( U ) ( s , t ; x ) = s Φ ( U ) ( s , t ; x ) ,
Φ t ( U ) ( s , t ; x ) = t Φ ( U ) ( s , t ; x ) .
5: 36.6 Scaling Relations
§36.6 Scaling Relations
Diffraction Catastrophe Scaling
Ψ K ( x ; k ) = k β K Ψ K ( y ( k ) ) ,
Ψ ( U ) ( x ; k ) = k β ( U ) Ψ ( U ) ( y ( U ) ( k ) ) ,
Table 36.6.1: Special cases of scaling exponents for cuspoids.
singularity K β K γ 1 K γ 2 K γ 3 K γ K
fold 1 1 6 2 3 - - 2 3