View source for Thom catastrophes

[[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).

<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>

</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 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" />.

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" />).

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.

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

<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>

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.

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" />.

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/t092650a.gif" />

Figure: t092650a

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/t092650b.gif" />

Figure: t092650b

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/t092650c.gif" />

Figure: t092650c

====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>



====Comments====


====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>