Math 2.0

Global analysis, analysis on manifolds Catastrophe theory Theorem

Thom's classification of elementary catastrophes

Builds on Versal unfoldingCodimension of a singularityGerm of a smooth functionCritical pointCatastrophe theory

Statement

Let r≤4r \le 4, and consider smooth families V ⁣:ℝn×ℝr→ℝV \colon \R^n \times \R^r \to \R, (x,u)↦Vu(x)(x, u) \mapsto V_u(x), of functions of nn state variables depending on rr control parameters. Give C∞(ℝn×ℝr,ℝ)C^\infty(\R^n \times \R^r, \R) the Whitney C∞C^\infty topology. There is an open dense set of families VV with the following property at every point (x0,u0)(x_0, u_0). View the germ of VV at (x0,u0)(x_0, u_0) as an unfolding of the germ of Vu0V_{u_0} at x0x_0. This unfolding is equivalent to a nondegenerate quadratic form in some state variables plus one of the following:

  1. a regular point: xx;
  2. a nondegenerate critical point: the function 00, so that only the quadratic form remains;
  3. the fold: x3+axx^3 + a x;
  4. the cusp: ±(x4+ax2+bx)\pm(x^4 + a x^2 + b x);
  5. the swallowtail: x5+ax3+bx2+cxx^5 + a x^3 + b x^2 + c x;
  6. the butterfly: ±(x6+ax4+bx3+cx2+dx)\pm(x^6 + a x^4 + b x^3 + c x^2 + d x);
  7. the hyperbolic umbilic: x3+y3+axy+bx+cyx^3 + y^3 + a xy + b x + c y;
  8. the elliptic umbilic: x3−3xy2+a(x2+y2)+bx+cyx^3 - 3 x y^2 + a (x^2 + y^2) + b x + c y;
  9. the parabolic umbilic: ±(x2y+y4+ax2+by2+cx+dy)\pm(x^2 y + y^4 + a x^2 + b y^2 + c x + d y).

Here equivalence of unfoldings allows a change of control parameters and a change of state variables depending on the controls. It also allows the addition of a smooth function of the controls. The letters a,b,c,da, b, c, d stand for the control parameters that are used.

Remarks

Items 3 to 9 are the seven elementary catastrophes. Each is the universal unfolding of its organising centre, the germ obtained by setting the parameters to 00. The number of parameters is the codimension of that germ. It is 11 for the fold, 22 for the cusp, 33 for the swallowtail and the hyperbolic and elliptic umbilics, and 44 for the butterfly and the parabolic umbilic. The idea of the proof has two parts. By Thom's transversality theorem in jet space, germs of codimension greater than rr do not occur in a generic rr-parameter family. Up to sign and quadratic terms, the germs of codimension from 11 to 44 are exactly the seven organising centres. These are x3x^3, x4x^4, x5x^5, x6x^6, x3+y3x^3 + y^3, x3−3xy2x^3 - 3xy^2 and x2y+y4x^2 y + y^4. Each of them is finitely determined, and its universal unfolding is checked with the versality criterion. The list of seven depends on the bound r≤4r \le 4: with five controls further catastrophes appear. For r≥6r \ge 6, families of germs with a modulus, such as x3+y3+z3+axyzx^3 + y^3 + z^3 + a xyz, occur generically, and no finite list is possible. Thom proposed the classification; complete proofs rest on the work of Mather on stability and finite determinacy.

Going deeper

For a student

The theorem combines the three main tools of local singularity theory: finite determinacy, the classification of germs of low codimension, and versal unfoldings. Its organising centres are the real forms of the simple singularities A2A_2 to A5A_5, D4±D_4^{\pm} and D5D_5 from Arnold's classification. The fold catastrophe and the cusp catastrophe are the two cases with one state variable and at most two controls, and they model sudden jumps and hysteresis.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Catastrophe theory, concept, no evidence levelCatastrophe theoryconceptCodimension of a singularity, concept, no evidence levelCodimension of a sing…conceptCritical point, concept, no evidence levelCritical pointconceptGerm of a smooth function, concept, no evidence levelGerm of a smooth func…conceptVersal unfolding, concept, no evidence levelVersal unfoldingconceptThom's classification of elementary catastrophes, statement, no evidence levelThom's classification…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Catastrophe theory concept
  2. Codimension of a singularity concept
  3. Critical point concept
  4. Germ of a smooth function concept
  5. Versal unfolding concept
  6. Thom's classification of elementary catastrophes statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Catastrophe_theory
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AboutCatastrophe theory, Versal unfolding
UsesVersal unfolding, Codimension of a singularity, Germ of a smooth function, Critical point, Catastrophe theory

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