Thom catastrophes

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

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