Lecture
Consider the germ of the mapping (9.3) at the point 0, assuming that
го .7;(0) = 1 and one of the conditions ЁЕ,у(0) # 0, Р,х»(0) = 0 is violated.
In this case any restrictions of inequality type may be imposed on all the other derivatives at the point 0.
There are two cases:
Е,(0) =0, Е„„(0) =0, Е (0) =0,
Ета (0) та 0, НЕ, (0) га 0;
Е, (0) =0, Е,.(0) =0, Е, „„(0) =0,
Ру(0) Е 0, Рот (0) 7 0,
where Н[ | denotes the Hessian of the function {.
In case (9.19) the point 0 is a nondegenerate critical point of the function А, (т, у), and by Morse's lemma, the germ of this function is С®-equivalent
to the quadratic form 12 + у”. For different signs the germs of the corresponding mappings are inequivalent even topologically: if
НЕ, (0) > 0, then the set of critical points 5 consists of a single
isolated point 0, whereas for Н[Е,|(0) < 0 it represents a
pair of curves intersecting transversally at 0.
In case (9.20) the set 5 is a smooth curve having, at
the point 0, second-order tangency with the field of kernels of the differential АХ”.
(9.19)
(9.20)
PROBLEM 9-5. Show that any germ of the mapping (9.3),
satisfying conditions (9.19) or (9.20), is С®-equivalent to a germ
of the form (9.3) with the function
Е(т, у) = 13 Е ту? + ф(х,у), фе, (9.21)
Е(т,у) = ту+ =‘ + ф(т,у), фЕм, (9.22)
respectively. Recall that м” denotes the ideal (in the algebra of germs of
functions) consisting of germs that are &-flat at the point 0.
SOLUTION. From the representation Р(т,у) = В (у) + 19(т,у) it follows
that by means of the left change 2 +} 2— 20 (1) one can eliminate all
monomials of the form у” in any jet of the function Р(т,у). Thus, in
case (9.19) we have
Р(х, у) = а? + 622 у + сту? + 9(2,у), а#0, рем.
Further, by means of a suitable right change т ++} ат + Ву one can
make а =1иф = 0. And with the help of a pair of changes у ++ \/|с]у, ш $ де]
we bring the germ of the mapping to the form (9.3) with the function (9.21).
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In case (9.20), after scaling the coordinate axes in the image and preimage, we have
Е(т, у) = ту+ т(аху + 6/7) + х(23 + ал?у + соту? + сзу3) + о(т,у,
where р Е м“. By means of the right change 17 ++ 1 + ах? + фту we kill
the cubic terms (а = В = 0), without changing at the same time the coefficients
of the monomials ту and т“. Then, by means of the change х +} ф- Ву one can
make с1 = 0. It is true that in doing so a new monomial appears — 82, but it
is killed by means of the left change 5+} 2-+ Вш?. As a result we obtain
Е(т, у) тут“ + сот? у? + злу? + ф(т,у), фм.
To bring the germ of the mapping to the form (9.3) with the function (9.22)
it remains only to achieve с5 = сз = 0, while preserving the coefficients of the monomials ту and 2“. This, obviously, is accomplished by means of the
left change 2 +} 2 -— 6227 — 63210. [|
Problem 9.5 is a preliminary for obtaining a stronger result:
PROBLEM 9.6. Show that the statement of problem 9.5 remains
valid if in formulas (9.21), (9.22) we set ф = 0.
HINT. The proof, with minor modifications, repeats the proof of theorem 9.1 for the cusp. The derivation of these normal forms, as well as the С®-normal forms for singularities of mappings of a plane to a plane of corank 1 and codimension greater than 3,
can also be found in the papers [36, 37].
The list of three normal forms given in problem 9.6,
=13-+ту, ш=у — lips (English: lips)
= — ту”, ш=у — beak (English: beak)
2=ху+1“, ш=у — swallowtail (English: swallowtail)
exhausts the list of singularities of mappings of a plane to a plane of codimension 3. Like the fold and the cusp, these mappings
can be realized as projections of surfaces onto a plane.
Unlike the fold and the cusp, these three singularities occur in a stable
manner (i.e. are not removed by small perturbations) not
for individual mappings of a plane to a plane, but for families
of such mappings, depending on at least one real parameter. The five mentioned singularities of codimension < 3 (fold,
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cusp, lips, beak, swallowtail) completely exhausts the list
of stable singularities of one-parameter families of mappings of a plane to a plane. The strange names «lips»,
«beak» and «swallowtail» are explained by the pictures that arise
in such families (see fig. 9.4, 9.5, 9.6 below).
EXAMPLE 9.2. Consider the family of mappings
2=Р(т,у,=) = + ЕД? ху, Ш=у, (9.23)
where = is a real parameter, realized as the projection of the surface 2 = Р(т,у.=Е) onto the plane (т,у) along the 2 axis. The critical points fill out the cubic parabola 5, given by the equation
Е, (т, у, Е) = 413 + 2=х + у = 0, at every point of which the field of kernels of the differential А’ is parallel to the 1 axis. At those points where the field А’ is transversal to 5, the projection mapping has a fold (see problem 9.2).
The following three cases differ substantially.
If = < 0, then the field ХА is tangent to the cubic parabola 5 at two
distinct points; at both the tangency is of first order and, consequently, at these points the projection has a singularity of cusp type;
see fig. 9.4 (top row, left). As = approaches zero from the left, the cusp points approach one another, and at = = 0 they merge
together at the origin, where the tangency of А and 5 is now of not first, but
second order. At Е = 0 the projection has, at the origin,
the «swallowtail» singularity; see fig. 9.4 (top row, center).
Finally, for = > 0 the field А’ is not tangent to 5 at any point, and all critical points are folds of the projection; see fig. 9.4 (top row,
right).
REMARK 9.4. Let us perturb the family (9.23) so that the «swallowtail» singularity at = = 0 disappears. For this it suffices to add to the function Р the term отЗ with any @ > 0. It is easy
to see that the «swallowtail» singularity arises at the parameter value
= = за?. One can show (the reader is advised to do this independently) that the same situation holds for any sufficiently small perturbation of the family (9.23). This is precisely what
the stability of this singularity in the family means.
REMARK 9.5. A change in the qualitative characteristics of objects of any nature, depending on a parameter, is called a «perestroika» (restructuring) or «bifurcation». The term «bifurcation» goes back to the great
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French mathematician Poincaré, who used it in studying dynamical systems (ordinary differential
equations) depending on a parameter. Nowadays it is most often used specifically with regard to dynamical systems. The term «perestroika», which came into use at the end of the 1980s,
is applied to objects of any nature”. Above we dealt with a perestroika of a family of mappings of a plane to a plane. =]
Fig. 9.4 Perestroika of the family of mappings (9.23). Left to right: = < 0,
= = 0, = > 0. Top row — preimage plane (т, у), the bold line —
the criminant, the dashed lines — the field А. Bottom row — image plane
(=, и), the bold line is the discriminant curve
EXAMPLE 9.3. Consider two families of mappings:
2 =Е(щ,у,=) = 13 + ту’+=т, ш=у, (9.24)
=Р(т,у,Е) = 13 -ту + Ех, ш=у. (9.25)
The criminant 5 of the mapping (9.24), given by the equation
За? Ну’ +Е=О,
for = < 0 is an ellipse centered at the origin of the
preimage plane. At all points of this ellipse except two, the field
р. In the English-language mathematical literature on singularity theory the transliteration «perestroika» is often used.
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А’ is transversal to 5, and the mapping has a fold. At two diametrically opposite points of the ellipse А’ is tangent to 5, and the mapping has a cusp. The discriminant curve of the mapping (9.24) has
two cusps and resembles slightly parted lips; see fig. 9.5 (left and center). As = tends to zero from the left, the ellipse (in the preimage) and the «lips»
(in the image) contract to a single point — the origin of the plane;
see fig. 9.5 (right). Then, for = > 0 the mapping (9.24) has no
critical points at all. p)
ый
/-1—\- __
$
у
А
% =
Ай
р
ыы =
Fig. 9.5 Perestroika of the family of mappings (9.24). Left and center:
Е < 0, right = = 0. Top row — preimage plane (т,у), the bold
line — the criminant, the dashed lines — the field А. Bottom row —
image plane (2,1), the bold line — the discriminant curve
The criminant 5 of the mapping (9.25), given by the equation
312 -Р+ЕЗО,
for all = = 0 is a hyperbola centered at the origin of the
preimage plane. The difference between = < и = >> 0 is that
in the first case the field Х is transversal to the branches of the hyperbola at
all points, the mapping has a fold, and the discriminant curve
consists of two smooth non-intersecting curves; see fig. 9.6 (left). In the second case the field А’ is transversal to the branches of the hyperbola at
all points except two, at which the mapping has a cusp. The discriminant curve consists of two non-intersecting curves resembling
semicubical parabolas; see fig. 9.6 (right). At = = 0
the criminant is a pair of intersecting lines — the asymptotic
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lines of the hyperbolas 352 — у? + Е = 0, and the discriminant curve is a pair of smooth curves touching each other at the origin; see
fig. 9.6 (center). < Хх
в и ы г < + ><
Fig. 9.6 Perestroika of the family of mappings (9.25). Left to right: = < 0,
Е = 0, Е > 0. Top row — preimage plane (т, у), the bold line —
the criminant, the dashed lines — the field Х. Bottom row — image plane
АХ | Ах
|
1
| |
(2, м), the bold line — the discriminant curve
PROBLEM 9.7. Show that all three singularities «lips», «beak»
and «swallowtail» are stable in one-parameter families of
mappings, i.e. do not disappear under small perturbations, but only
«move» from one parameter value to another.
PROBLEM 9.8. Investigate the perestroikas of the families of mappings
д = Р(т, у) +=ф(т,у,=), ш=у,
where Ё = 21 + ху (swallowtail), Г = 23 + ту? (lips and beak), ф is
an arbitrary smooth function, Е is a small parameter. Show that
under some general condition on ф such perestroikas are
analogous to those described above in examples 9.2 and 9.3.
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