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[[Singularities of differentiable mappings|Singularities of differentiable mappings]], whose classification was announced by R. Thom [[#References|[1]]] in terms of their gradient dynamical systems and the analogous list of critical points of codimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t0926501.png" /> (cf. [[Critical point|Critical point]]) of differentiable functions. The original formulation of Thom's result is that a generic four-parameter family of functions is stable, and in the neighbourhood of a critical point it behaves, up to sign and change of variable, like one of seven cases (cf. Table).
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<table border="0" cellpadding="0" cellspacing="0" style="background-color:black;"> <tr><td> <table border="0" cellspacing="1" cellpadding="4" style="background-color:black;"> <tbody> <tr> <td colname="1" style="background-color:white;" colspan="1">Notation</td> <td colname="2" style="background-color:white;" colspan="1">Codim</td> <td colname="3" style="background-color:white;" colspan="1">Corank</td> <td colname="4" style="background-color:white;" colspan="1">Germ</td> <td colname="5" style="background-color:white;" colspan="1">Universal deformation</td> <td colname="6" style="background-color:white;" colspan="1">Name</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t0926502.png" /></td> <td colname="2" style="background-color:white;" colspan="1">1</td> <td colname="3" style="background-color:white;" colspan="1">1</td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t0926503.png" /></td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t0926504.png" /></td> <td colname="6" style="background-color:white;" colspan="1">Fold</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t0926505.png" /></td> <td colname="2" style="background-color:white;" colspan="1">2</td> <td colname="3" style="background-color:white;" colspan="1">1</td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t0926506.png" /></td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t0926507.png" /></td> <td colname="6" style="background-color:white;" colspan="1">Cusp</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t0926508.png" /></td> <td colname="2" style="background-color:white;" colspan="1">3</td> <td colname="3" style="background-color:white;" colspan="1">1</td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t0926509.png" /></td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265010.png" /></td> <td colname="6" style="background-color:white;" colspan="1">Swallow-tail</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265011.png" /></td> <td colname="2" style="background-color:white;" colspan="1">3</td> <td colname="3" style="background-color:white;" colspan="1">2</td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265012.png" /></td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265013.png" /></td> <td colname="6" style="background-color:white;" colspan="1">Hyperbolic umbilic</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265014.png" /></td> <td colname="2" style="background-color:white;" colspan="1">3</td> <td colname="3" style="background-color:white;" colspan="1">2</td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265015.png" /></td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265016.png" /></td> <td colname="6" style="background-color:white;" colspan="1">Elliptic umbilic</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265017.png" /></td> <td colname="2" style="background-color:white;" colspan="1">4</td> <td colname="3" style="background-color:white;" colspan="1">2</td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265018.png" /></td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265019.png" /></td> <td colname="6" style="background-color:white;" colspan="1">Butterfly</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265020.png" /></td> <td colname="2" style="background-color:white;" colspan="1">4</td> <td colname="3" style="background-color:white;" colspan="1">2</td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265021.png" /></td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265022.png" /></td> <td colname="6" style="background-color:white;" colspan="1">Parabolic umbilic</td> </tr> </tbody> </table>
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[[Singularities of differentiable mappings|Singularities of differentiable mappings]], whose classification was announced by R. Thom [[#References|[1]]] in terms of their gradient dynamical systems and the analogous list of critical points of codimension  $  \leq  4 $(
 +
cf. [[Critical point|Critical point]]) of differentiable functions. The original formulation of Thom's result is that a generic four-parameter family of functions is stable, and in the neighbourhood of a critical point it behaves, up to sign and change of variable, like one of seven cases (cf. Table).
 +
 
 +
<table border="0" cellpadding="0" cellspacing="0" style="background-color:black;"> <tr><td> <table border="0" cellspacing="1" cellpadding="4" style="background-color:black;"> <tbody> <tr> <td colname="1" style="background-color:white;" colspan="1">Notation</td> <td colname="2" style="background-color:white;" colspan="1">Codim</td> <td colname="3" style="background-color:white;" colspan="1">Corank</td> <td colname="4" style="background-color:white;" colspan="1">Germ</td> <td colname="5" style="background-color:white;" colspan="1">Universal deformation</td> <td colname="6" style="background-color:white;" colspan="1">Name</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  A _ {2} $
 +
</td> <td colname="2" style="background-color:white;" colspan="1">1</td> <td colname="3" style="background-color:white;" colspan="1">1</td> <td colname="4" style="background-color:white;" colspan="1"> $  x  ^ {3} + y  ^ {2} $
 +
</td> <td colname="5" style="background-color:white;" colspan="1"> $  ux $
 +
</td> <td colname="6" style="background-color:white;" colspan="1">Fold</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  A _ {3} $
 +
</td> <td colname="2" style="background-color:white;" colspan="1">2</td> <td colname="3" style="background-color:white;" colspan="1">1</td> <td colname="4" style="background-color:white;" colspan="1"> $  x  ^ {4} + y  ^ {2} $
 +
</td> <td colname="5" style="background-color:white;" colspan="1"> $  ux + vx  ^ {2} $
 +
</td> <td colname="6" style="background-color:white;" colspan="1">Cusp</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  A _ {4} $
 +
</td> <td colname="2" style="background-color:white;" colspan="1">3</td> <td colname="3" style="background-color:white;" colspan="1">1</td> <td colname="4" style="background-color:white;" colspan="1"> $  x  ^ {5} + y  ^ {2} $
 +
</td> <td colname="5" style="background-color:white;" colspan="1"> $  ux + vx  ^ {2} + ux  ^ {3} $
 +
</td> <td colname="6" style="background-color:white;" colspan="1">Swallow-tail</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  D _ {4}  ^ {-} $
 +
</td> <td colname="2" style="background-color:white;" colspan="1">3</td> <td colname="3" style="background-color:white;" colspan="1">2</td> <td colname="4" style="background-color:white;" colspan="1"> $  x  ^ {3} + xy  ^ {2} $
 +
</td> <td colname="5" style="background-color:white;" colspan="1"> $  ux + vx  ^ {2} + wy $
 +
</td> <td colname="6" style="background-color:white;" colspan="1">Hyperbolic umbilic</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  D _ {4}  ^ {+} $
 +
</td> <td colname="2" style="background-color:white;" colspan="1">3</td> <td colname="3" style="background-color:white;" colspan="1">2</td> <td colname="4" style="background-color:white;" colspan="1"> $  x  ^ {3-} xy  ^ {2} $
 +
</td> <td colname="5" style="background-color:white;" colspan="1"> $  ux + vx  ^ {2} + wy $
 +
</td> <td colname="6" style="background-color:white;" colspan="1">Elliptic umbilic</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  A _ {5} $
 +
</td> <td colname="2" style="background-color:white;" colspan="1">4</td> <td colname="3" style="background-color:white;" colspan="1">2</td> <td colname="4" style="background-color:white;" colspan="1"> $  x  ^ {6} + y  ^ {2} $
 +
</td> <td colname="5" style="background-color:white;" colspan="1"> $  ux + vx  ^ {2} + wx  ^ {3} + tx  ^ {4} $
 +
</td> <td colname="6" style="background-color:white;" colspan="1">Butterfly</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  D _ {5} $
 +
</td> <td colname="2" style="background-color:white;" colspan="1">4</td> <td colname="3" style="background-color:white;" colspan="1">2</td> <td colname="4" style="background-color:white;" colspan="1"> $  x  ^ {4} + xy  ^ {2} $
 +
</td> <td colname="5" style="background-color:white;" colspan="1"> $  ux + vx  ^ {2} + wx  ^ {3} + ty $
 +
</td> <td colname="6" style="background-color:white;" colspan="1">Parabolic umbilic</td> </tr> </tbody> </table>
  
 
</td></tr> </table>
 
</td></tr> </table>
  
The germs (cf. [[Germ|Germ]]) corresponding to the Thom catastrophes are finitely determined (specifically, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265023.png" />-determined: in appropriate coordinates they correspond to polynomials in two variables of degrees <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265024.png" />).
+
The germs (cf. [[Germ|Germ]]) corresponding to the Thom catastrophes are finitely determined (specifically, $  6 $-
 +
determined: in appropriate coordinates they correspond to polynomials in two variables of degrees $  \leq  6 $).
  
The codimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265025.png" /> serves as a measure of the complexity of a critical point. Any small perturbation of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265026.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265027.png" /> leads to a function with at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265028.png" /> complex critical points. The codimension of a singularity (that is, of a germ <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265029.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265030.png" />) is the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265031.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265033.png" /> is the ideal generated by the germs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265034.png" />. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265035.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265036.png" />, and the residue classes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265037.png" /> form a basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265038.png" />, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265039.png" />. The inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265040.png" />, holds, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265041.png" /> is the corank of the Hessian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265042.png" />. Hence, in particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265043.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265044.png" />.
+
The codimension $  \mathop{\rm codim} $
 +
serves as a measure of the complexity of a critical point. Any small perturbation of a function $  f $
 +
of $  \mathop{\rm codim}  r $
 +
leads to a function with at most $  r $
 +
complex critical points. The codimension of a singularity (that is, of a germ $  f $
 +
such that $  f ( 0) = Df ( 0) = 0 $)  
 +
is the number $  \mathop{\rm dim}  \mathfrak m /\langle  \partial  f \rangle $,  
 +
where $  \mathfrak m = \{ {g } : {g( 0) = 0 } \} $
 +
and $  \langle  \partial  f \rangle $
 +
is the ideal generated by the germs $  \partial  f/ \partial  x  ^ {i} $.  
 +
For example, if $  f = x  ^ {N} $,  
 +
then $  \langle  \partial  f \rangle = \langle  x ^ {N - 1 } \rangle $,  
 +
and the residue classes of $  x \dots x ^ {N - 2 } $
 +
form a basis of $  \mathfrak m /\langle  \partial  f \rangle $,  
 +
so that $  \mathop{\rm codim} = N- 2 $.  
 +
The inequality $  \mathop{\rm codim}  f \geq  c ( c + 1)/2 $,  
 +
holds, where $  c $
 +
is the corank of the Hessian $  \partial  ^ {2} f/ \partial  x  ^ {i} \partial  x  ^ {j} ( 0) $.  
 +
Hence, in particular, if $  r \leq  4 $,  
 +
then $  c \leq  2 $.
  
Finite determination (sufficiency) of a germ, roughly speaking, means that it is determined, up to smooth change of coordinates, by its jets (cf. [[Jet|Jet]]). More precisely, a germ <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265045.png" /> is said to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265047.png" />-determined if every germ <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265048.png" /> with the same <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265049.png" />-jet (that is, the same Taylor series up to order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265050.png" />) as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265051.png" /> is right equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265052.png" /> (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265053.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265054.png" /> is the germ at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265055.png" /> of a diffeomorphism; cf. [[#References|[2]]]). A germ is finitely determined if and only if it has finite codimension. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265056.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265057.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265058.png" />-determined (whence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265059.png" />-determined for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265060.png" />).
+
Finite determination (sufficiency) of a germ, roughly speaking, means that it is determined, up to smooth change of coordinates, by its jets (cf. [[Jet|Jet]]). More precisely, a germ $  f $
 +
is said to be $  k $-
 +
determined if every germ $  f _ {1} $
 +
with the same $  k $-
 +
jet (that is, the same Taylor series up to order $  k $)  
 +
as $  f $
 +
is right equivalent to $  f $(
 +
i.e. $  f _ {1} = f \circ \varphi $
 +
where $  \varphi $
 +
is the germ at 0 $
 +
of a diffeomorphism; cf. [[#References|[2]]]). A germ is finitely determined if and only if it has finite codimension. In particular, if $  \mathop{\rm codim} = r $,  
 +
then $  f $
 +
is $  ( r + 2) $-
 +
determined (whence $  6 $-
 +
determined for $  r \leq  4 $).
  
The Thom catastrophes, in contrast to the case of [[General position|general position]], are degenerate singularities (that is, the Hessian is degenerate at them), and they can be removed by a small perturbation, as mentioned above. However, in many cases of practical importance, and also theoretically, one is interested not in an individual object, but in a collection of them, depending on some "control" parameters. Degenerate singularities which are removable for each fixed value of the parameters may be removable for the collection as a whole. (Stability of Thom catastrophes may also be considered in this sense.) But then the natural object of study is not the singularity itself, but a collection (a deformation of the singularity) in which it is non-removable (or disintegrates, or "bifurcates" ) under a change of parameters. It turns out that in many cases the study of all possible deformations can be reduced to the study of a single one, which is in a certain sense so big that all the others can be obtained from it. Such deformations are called versal and they, in turn, can be obtained from a universal (or miniversal) deformation, which is characterized by having least possible dimension of its parameter space. The most important result here is Mather's theorem: A singularity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265061.png" /> has a universal deformation if and only if its codimension is finite.
+
The Thom catastrophes, in contrast to the case of [[General position|general position]], are degenerate singularities (that is, the Hessian is degenerate at them), and they can be removed by a small perturbation, as mentioned above. However, in many cases of practical importance, and also theoretically, one is interested not in an individual object, but in a collection of them, depending on some "control" parameters. Degenerate singularities which are removable for each fixed value of the parameters may be removable for the collection as a whole. (Stability of Thom catastrophes may also be considered in this sense.) But then the natural object of study is not the singularity itself, but a collection (a deformation of the singularity) in which it is non-removable (or disintegrates, or "bifurcates" ) under a change of parameters. It turns out that in many cases the study of all possible deformations can be reduced to the study of a single one, which is in a certain sense so big that all the others can be obtained from it. Such deformations are called versal and they, in turn, can be obtained from a universal (or miniversal) deformation, which is characterized by having least possible dimension of its parameter space. The most important result here is Mather's theorem: A singularity $  f $
 +
has a universal deformation if and only if its codimension is finite.
  
A deformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265062.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265063.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265064.png" />, of a germ <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265065.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265066.png" />, is given by a formula
+
A deformation $  F ( x, u) $,  
 +
$  x \in \mathbf R  ^ {n} $,  
 +
$  u \in \mathbf R  ^ {r} $,  
 +
of a germ $  f ( x) $,
 +
$  F ( x, 0) = f ( x) $,  
 +
is given by a formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265067.png" /></td> </tr></table>
+
$$
 +
F ( x, u)  = \
 +
f ( x) + b _ {1} u _ {1} + \dots + b _ {r} u _ {r} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265068.png" /> is an arbitrary collection of representative elements of a basis of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265069.png" />. Thom catastrophes correspond to deformations with at most four parameters.
+
where $  ( b _ {1} \dots b _ {r} ) $
 +
is an arbitrary collection of representative elements of a basis of the space $  \mathfrak m /\langle  \partial  f \rangle $.  
 +
Thom catastrophes correspond to deformations with at most four parameters.
  
Important for applications is the so-called bifurcation set, or singular set, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265070.png" />; its projection to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265071.png" />-space, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265072.png" />, is called the catastrophe set. It lies in the control space and hence is "observable" , and all "discontinuities" or "catastrophes" originate from it. Fig.1a, Fig.1b and Fig.1c illustrate the cases corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092650/t09265073.png" />.
+
Important for applications is the so-called bifurcation set, or singular set, $  D _ {f} = \{ {( x, u) \in \mathbf R  ^ {n} \times U } : {d _ {x} f = 0 \textrm{ and }  d  ^ {2} f  \textrm{ is  degenerate  } } \} $;  
 +
its projection to the $  u $-
 +
space, the set $  \{ {u \in U } : {( x, u) \in D _ {f}  \textrm{ for  some  }  x \in \mathbf R  ^ {n} } \} $,  
 +
is called the catastrophe set. It lies in the control space and hence is "observable" , and all "discontinuities" or "catastrophes" originate from it. Fig.1a, Fig.1b and Fig.1c illustrate the cases corresponding to $  \mathop{\rm codim}  3 $.
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/t092650a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/t092650a.gif" />
Line 35: Line 117:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> R. Thom, "Topological models in biology" ''Topology'' , '''8''' (1969) pp. 313–335 {{MR|0245318}} {{ZBL|0165.23301}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> P. Bröcker, L. Lander, "Differentiable germs and catastrophes" , Cambridge Univ. Press (1975) {{MR|0494220}} {{ZBL|0302.58006}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> T. Poston, I. Stewart, "Catastrophe theory and its applications" , Pitman (1978) {{MR|0501079}} {{ZBL|0382.58006}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> R. Thom, "Topological models in biology" ''Topology'' , '''8''' (1969) pp. 313–335 {{MR|0245318}} {{ZBL|0165.23301}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> P. Bröcker, L. Lander, "Differentiable germs and catastrophes" , Cambridge Univ. Press (1975) {{MR|0494220}} {{ZBL|0302.58006}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> T. Poston, I. Stewart, "Catastrophe theory and its applications" , Pitman (1978) {{MR|0501079}} {{ZBL|0382.58006}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Thom, "Structural stability and morphogenesis" , Benjamin (1976) (Translated from French) {{MR|0488155}} {{MR|0488156}} {{ZBL|0392.92001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R. Thom, "Mathematical models of morphogenesis" , Wiley (1983) (Translated from French) {{MR|0729829}} {{ZBL|0565.92002}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> V.I. Arnol'd, "Catastrophe theory" , Springer (1984) (Translated from Russian) {{MR|}} {{ZBL|0791.00009}} {{ZBL|0746.58001}} {{ZBL|0704.58001}} {{ZBL|0721.01001}} {{ZBL|0674.01033}} {{ZBL|0645.58001}} {{ZBL|0797.58002}} {{ZBL|0517.58002}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> E.C. Zeeman, "Catastrophe theory" , Addison-Wesley (1977) {{MR|0474383}} {{ZBL|0398.58012}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Thom, "Structural stability and morphogenesis" , Benjamin (1976) (Translated from French) {{MR|0488155}} {{MR|0488156}} {{ZBL|0392.92001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R. Thom, "Mathematical models of morphogenesis" , Wiley (1983) (Translated from French) {{MR|0729829}} {{ZBL|0565.92002}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> V.I. Arnol'd, "Catastrophe theory" , Springer (1984) (Translated from Russian) {{MR|}} {{ZBL|0791.00009}} {{ZBL|0746.58001}} {{ZBL|0704.58001}} {{ZBL|0721.01001}} {{ZBL|0674.01033}} {{ZBL|0645.58001}} {{ZBL|0797.58002}} {{ZBL|0517.58002}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> E.C. Zeeman, "Catastrophe theory" , Addison-Wesley (1977) {{MR|0474383}} {{ZBL|0398.58012}} </TD></TR></table>

Revision as of 08:25, 6 June 2020


Singularities of differentiable mappings, whose classification was announced by R. Thom [1] in terms of their gradient dynamical systems and the analogous list of critical points of codimension $ \leq 4 $( cf. Critical point) of differentiable functions. The original formulation of Thom's result is that a generic four-parameter family of functions is stable, and in the neighbourhood of a critical point it behaves, up to sign and change of variable, like one of seven cases (cf. Table).

<tbody> </tbody>
Notation Codim Corank Germ Universal deformation Name
$ A _ {2} $ 1 1 $ x ^ {3} + y ^ {2} $ $ ux $ Fold
$ A _ {3} $ 2 1 $ x ^ {4} + y ^ {2} $ $ ux + vx ^ {2} $ Cusp
$ A _ {4} $ 3 1 $ x ^ {5} + y ^ {2} $ $ ux + vx ^ {2} + ux ^ {3} $ Swallow-tail
$ D _ {4} ^ {-} $ 3 2 $ x ^ {3} + xy ^ {2} $ $ ux + vx ^ {2} + wy $ Hyperbolic umbilic
$ D _ {4} ^ {+} $ 3 2 $ x ^ {3-} xy ^ {2} $ $ ux + vx ^ {2} + wy $ Elliptic umbilic
$ A _ {5} $ 4 2 $ x ^ {6} + y ^ {2} $ $ ux + vx ^ {2} + wx ^ {3} + tx ^ {4} $ Butterfly
$ D _ {5} $ 4 2 $ x ^ {4} + xy ^ {2} $ $ ux + vx ^ {2} + wx ^ {3} + ty $ Parabolic umbilic

The germs (cf. Germ) corresponding to the Thom catastrophes are finitely determined (specifically, $ 6 $- determined: in appropriate coordinates they correspond to polynomials in two variables of degrees $ \leq 6 $).

The codimension $ \mathop{\rm codim} $ serves as a measure of the complexity of a critical point. Any small perturbation of a function $ f $ of $ \mathop{\rm codim} r $ leads to a function with at most $ r $ complex critical points. The codimension of a singularity (that is, of a germ $ f $ such that $ f ( 0) = Df ( 0) = 0 $) is the number $ \mathop{\rm dim} \mathfrak m /\langle \partial f \rangle $, where $ \mathfrak m = \{ {g } : {g( 0) = 0 } \} $ and $ \langle \partial f \rangle $ is the ideal generated by the germs $ \partial f/ \partial x ^ {i} $. For example, if $ f = x ^ {N} $, then $ \langle \partial f \rangle = \langle x ^ {N - 1 } \rangle $, and the residue classes of $ x \dots x ^ {N - 2 } $ form a basis of $ \mathfrak m /\langle \partial f \rangle $, so that $ \mathop{\rm codim} = N- 2 $. The inequality $ \mathop{\rm codim} f \geq c ( c + 1)/2 $, holds, where $ c $ is the corank of the Hessian $ \partial ^ {2} f/ \partial x ^ {i} \partial x ^ {j} ( 0) $. Hence, in particular, if $ r \leq 4 $, then $ c \leq 2 $.

Finite determination (sufficiency) of a germ, roughly speaking, means that it is determined, up to smooth change of coordinates, by its jets (cf. Jet). More precisely, a germ $ f $ is said to be $ k $- determined if every germ $ f _ {1} $ with the same $ k $- jet (that is, the same Taylor series up to order $ k $) as $ f $ is right equivalent to $ f $( i.e. $ f _ {1} = f \circ \varphi $ where $ \varphi $ is the germ at $ 0 $ of a diffeomorphism; cf. [2]). A germ is finitely determined if and only if it has finite codimension. In particular, if $ \mathop{\rm codim} = r $, then $ f $ is $ ( r + 2) $- determined (whence $ 6 $- determined for $ r \leq 4 $).

The Thom catastrophes, in contrast to the case of general position, are degenerate singularities (that is, the Hessian is degenerate at them), and they can be removed by a small perturbation, as mentioned above. However, in many cases of practical importance, and also theoretically, one is interested not in an individual object, but in a collection of them, depending on some "control" parameters. Degenerate singularities which are removable for each fixed value of the parameters may be removable for the collection as a whole. (Stability of Thom catastrophes may also be considered in this sense.) But then the natural object of study is not the singularity itself, but a collection (a deformation of the singularity) in which it is non-removable (or disintegrates, or "bifurcates" ) under a change of parameters. It turns out that in many cases the study of all possible deformations can be reduced to the study of a single one, which is in a certain sense so big that all the others can be obtained from it. Such deformations are called versal and they, in turn, can be obtained from a universal (or miniversal) deformation, which is characterized by having least possible dimension of its parameter space. The most important result here is Mather's theorem: A singularity $ f $ has a universal deformation if and only if its codimension is finite.

A deformation $ F ( x, u) $, $ x \in \mathbf R ^ {n} $, $ u \in \mathbf R ^ {r} $, of a germ $ f ( x) $, $ F ( x, 0) = f ( x) $, is given by a formula

$$ F ( x, u) = \ f ( x) + b _ {1} u _ {1} + \dots + b _ {r} u _ {r} , $$

where $ ( b _ {1} \dots b _ {r} ) $ is an arbitrary collection of representative elements of a basis of the space $ \mathfrak m /\langle \partial f \rangle $. Thom catastrophes correspond to deformations with at most four parameters.

Important for applications is the so-called bifurcation set, or singular set, $ D _ {f} = \{ {( x, u) \in \mathbf R ^ {n} \times U } : {d _ {x} f = 0 \textrm{ and } d ^ {2} f \textrm{ is degenerate } } \} $; its projection to the $ u $- space, the set $ \{ {u \in U } : {( x, u) \in D _ {f} \textrm{ for some } x \in \mathbf R ^ {n} } \} $, is called the catastrophe set. It lies in the control space and hence is "observable" , and all "discontinuities" or "catastrophes" originate from it. Fig.1a, Fig.1b and Fig.1c illustrate the cases corresponding to $ \mathop{\rm codim} 3 $.

Figure: t092650a

Figure: t092650b

Figure: t092650c

References

[1] R. Thom, "Topological models in biology" Topology , 8 (1969) pp. 313–335 MR0245318 Zbl 0165.23301
[2] P. Bröcker, L. Lander, "Differentiable germs and catastrophes" , Cambridge Univ. Press (1975) MR0494220 Zbl 0302.58006
[3] T. Poston, I. Stewart, "Catastrophe theory and its applications" , Pitman (1978) MR0501079 Zbl 0382.58006

Comments

References

[a1] R. Thom, "Structural stability and morphogenesis" , Benjamin (1976) (Translated from French) MR0488155 MR0488156 Zbl 0392.92001
[a2] R. Thom, "Mathematical models of morphogenesis" , Wiley (1983) (Translated from French) MR0729829 Zbl 0565.92002
[a3] V.I. Arnol'd, "Catastrophe theory" , Springer (1984) (Translated from Russian) Zbl 0791.00009 Zbl 0746.58001 Zbl 0704.58001 Zbl 0721.01001 Zbl 0674.01033 Zbl 0645.58001 Zbl 0797.58002 Zbl 0517.58002
[a4] E.C. Zeeman, "Catastrophe theory" , Addison-Wesley (1977) MR0474383 Zbl 0398.58012
How to Cite This Entry:
Thom catastrophes. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Thom_catastrophes&oldid=48968
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article