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36 Integrals with Coalescing SaddlesProperties

§36.4 Bifurcation Sets

Contents

§36.4(i) Formulas

Critical Points for Cuspoids

These are real solutions tj(x), 1jjmax(x)K+1, of

36.4.1 tΦK(tj(x);x)=0.

Critical Points for Umbilics

These are real solutions {sj(x),tj(x)}, 1jjmax(x)4, of

36.4.2 sΦ(U)(sj(x),tj(x);x) =0,
tΦ(U)(sj(x),tj(x);x) =0.

Bifurcation (Catastrophe) Set for Cuspoids

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

36.4.3 2t2ΦK(t;x)=0.

Bifurcation (Catastrophe) Set for Umbilics

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

36.4.4 2s2Φ(U)(s,t;x)2t2Φ(U)(s,t;x)-(2stΦ(U)(s,t;x))2=0.

Special Cases

K=1, fold bifurcation set:

36.4.5 x=0.

K=2, cusp bifurcation set:

36.4.6 27x2=-8y3.

K=3, swallowtail bifurcation set:

36.4.7 x =3t2(z+5t2),
y =-t(3z+10t2),
-<t<.

Swallowtail self-intersection line:

36.4.8 y =0,
z 0,
x =920z2.

Swallowtail cusp lines (ribs):

36.4.9 z 0,
x =-320z2,
10y2 =-4z3.

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

36.4.10 x =13z2(-cos(2ϕ)-2cosϕ),
y =13z2(sin(2ϕ)-2sinϕ),
0ϕ2π.

Elliptic umbilic cusp lines (ribs):

36.4.11 x+iy=-z2exp(23iπm),
m=0,1,2.

Hyperbolic umbilic bifurcation set (codimension three):

36.4.12 x =-112z2(exp(2τ)±2exp(-τ)),
y =-112z2(exp(-2τ)±2exp(τ)),
-τ<.

The + sign labels the cusped sheet; the - sign labels the sheet that is smooth for z0 (see Figure 36.4.4).

Hyperbolic umbilic cusp line (rib):

36.4.13 x=y=-14z2.

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

§36.4(ii) Visualizations

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
See accompanying text
Figure 36.4.3: Bifurcation set of elliptic umbilic catastrophe. Magnify
See accompanying text
Figure 36.4.4: Bifurcation set of hyperbolic umbilic catastrophe. Magnify