Disaster theory (mathematics)

Mathematical catastrophe theory deals with discontinuous, abrupt changes in continuous dynamic systems . These can, even if they are seeking a stable state under certain conditions, sudden changes in the parameters, not continuous , discontinuous changes of the solution out.

The catastrophe theory examines the branching behavior of these solutions ( bifurcations ) when the parameters vary and is therefore an important basis for the mathematical treatment of chaos theory . Sometimes mathematics prefers to speak of the theory of the singularities of differentiable mappings , and the lurid name disaster theory is avoided. The main result is the division of these singularities into seven "normal types".

The catastrophe theory is fundamentally based on the differential topology . It was developed in the late 1960s by René Thom , Wladimir Arnold and others. It is used and expanded, among other things, in modern physics and economics , but also in linguistics and psychology and was therefore popular in these areas in the 1970s as a directly applicable qualitative mathematical method. The English mathematician Erik Christopher Zeeman , who applied the theory of ship stability to the theory of evolution , was particularly active . This also led to a backlash and criticism of the applications of the theory (specifically by Zeeman) from the 1970s onwards. As the title of his book from 1972 shows, Thom himself was primarily looking for applications in biology (especially embryo development, morphogenesis ).

Elementary disasters

The catastrophe theory analyzes degenerate critical points of potential functions. These are points where, in addition to all of the first derivatives, some of the higher derivatives are also zero. The points form the germ of the catastrophe geometries. The degeneracy can be "unfolded" by developing the potential function in a Taylor series and a small perturbation of the parameter.

If the critical points cannot be eliminated by small disturbances, they are called structurally stable. Their geometric structure can be classified with three or fewer variables of the potential function and five or fewer parameters of this function by only seven types of (bifurcation) geometries. They correspond to the normal forms to which the Taylor expansion around catastrophe germs can be traced back with the help of diffeomorphisms (differentiable maps).

Mathematical formulation

We consider potential functions in variables that depend on free parameters , i.e. differentiable functions . Let be the set of critical values, i.e. the solutions of , as a subset of . The projection onto the parameter space defines the "catastrophe mapping" . ${\displaystyle r}$ ${\displaystyle n}$ ${\displaystyle V\colon \mathbb {R} ^{n}\times \mathbb {R} ^{r}\to \mathbb {R} }$ ${\displaystyle M}$ ${\displaystyle {\frac {\partial V}{\partial x_{1}}}=\ldots ={\frac {\partial V}{\partial x_{n}}}=0}$ ${\displaystyle \mathbb {R} ^{n}\times \mathbb {R} ^{r}}$ ${\displaystyle \mathbb {R} ^{r}}$ ${\displaystyle X\colon M\to \mathbb {R} ^{r}}$

Thom's theorem says that for generic functions

• ${\displaystyle M}$is a -dimensional manifold, ${\displaystyle r}$
• each singularity of the catastrophe map is equivalent to one from the list of elementary catastrophes , ${\displaystyle X\colon M\to \mathbb {R} ^{r}}$
• the disaster map is locally stable at each point of with respect to small disturbances from . ${\displaystyle X}$ ${\displaystyle M}$ ${\displaystyle V}$

Potential functions of a variable

In practice, the folding bifurcations and the tip catastrophe (cusp geometry) are by far the most important cases in catastrophe theory and occur in numerous cases. The remaining catastrophes, however, are very specific and are only listed here for the sake of completeness.

Stable and unstable pairs of extremes disappear in a folding catastrophe

Folding disaster ( fold catastrophe )

${\displaystyle V=x^{3}+ax\,}$

With negative values ​​of a , the potential function has a stable and an unstable extremum. If the parameter a increases slowly, the system can follow the stable minimum. At a = 0, the stable and unstable extrema meet and cancel each other out (bifurcation point). For a > 0 there is no longer a stable solution. At a = 0, a physical system would suddenly lose the stability it had for negative a and its behavior overturned .

Acute disaster ( cusp catastrophe )

${\displaystyle V=x^{4}+ax^{2}+bx\,}$

The cusp disaster occurs quite often when considering the behavior of a convolution disaster when a second parameter b is added to the parameter space. If you now change the parameters, there is a curve (blue in the figure) of points in the parameter space (a, b) which, when exceeded, lose stability. Instead of one extreme, there are now two to which the system can jump. If one changes b periodically , one can create a "jumping back and forth" in the local space . However, this is only possible for the range a <0; the closer a approaches zero, the smaller the hysteresis curves become and finally disappear completely at a = 0.

Conversely, if one keeps b constant and varies a one observes a tuning fork bifurcation ( pitchfork bifurcation ) in the symmetrical case b = 0 : if a decreases, one stable solution suddenly splits into two stable and one unstable solution when the system breaks the cusp Point a = 0, b = 0 to negative values ​​of a happened. This is an example of a spontaneous break in symmetry . Further away from the cusp point, this sudden change in the structure of the solution is missing and only a second possible solution appears.

A well-known example uses the cusp to model the behavior of a stressed dog between submission and aggressiveness. With moderate stress ( a > 0), depending on the provocation (parameter b), the dog shows a steady transition behavior between the two behaviors. In the case of higher stress (region a <0 ), the dog remains intimidated even with a weakened provocation, only to suddenly tip over into aggressive behavior when the folding point is reached , which it maintains even when the provocation parameter is reduced.

Another example is the transition of a magnetic system (more precisely: a ferromagnetic system, e.g. iron ) at the critical point (a = b = 0) when the temperature falls below the critical temperature T _c from the non-magnetic to the ferromagnetic state. The parameter a is proportional to the temperature difference TT _c and b is proportional to the magnetic field. Furthermore, one can explain the term spontaneous symmetry breaking very well with this example , since in the ferromagnetic state - depending on the sign of a very weak symmetry breaking magnetic field - one of the two directions shown is preferred.

The catastrophe theory neglects the fluctuations occurring here in the vicinity of the critical point , e.g. B. the magnetization fluxes (cf. Monodromy ).

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  Diagram of a three-dimensional cusp catastrophe with curves (brown, red) of critical points in x with variation of the parameters a and b.
  Outside the cusps (blue) there is only one extremum, inside there are two.

• 

  Shape of the cusp in the parameter space (a, b) near a catastrophe point, which separates the areas of one and two extremes, for a special curve selection of a (x ² ) and b (x ² ).

• 

  Special case of a cusp catastrophe (“pitchfork bifurcation”) at a = 0 in the cut surface b = 0

Dovetail disaster ( swallowtail catastrophe )

The swallowtail disaster

${\displaystyle V=x^{5}+ax^{3}+bx^{2}+cx\,}$

Here the space of the control parameters is three-dimensional. The bifurcation set consists of three areas of folding catastrophes that meet in two cusp bifurcations. These in turn meet at a single dovetail bifurcation point.

If the parameters go through the areas of the folding bifurcations, a minimum and a maximum of the potential function vanish. At the cusp bifurcations, two minima and one maximum are replaced by a minimum, behind them the folding bifurcation disappears. In the dovetail point, two minima and two maxima meet at a single point x . For values a > 0, beyond the dovetail, there is either a maximum-minimum pair or none at all, depending on the parameter values b and c . Two of the surfaces of the folding bifurcations and the two curves of the cusp bifurcations disappear at the dovetail point and only a single surface of folding bifurcations remains. Salvador Dalí's last painting, The Swallowtail, was based on this catastrophe.

Butterfly disaster ( butterfly catastrophe )

${\displaystyle V=x^{6}+ax^{4}+bx^{3}+cx^{2}+dx\,}$

Depending on the parameters, the potential function can have 3, 2 or 1 local minimum. The different areas are separated in the parameter space by convolutional bifurcations. At the butterfly point, the different 3-faces of folding bifurcations, 2-faces of cusp bifurcations and curves of butterfly bifurcations meet and disappear to leave only a single cusp structure for a> 0 .

Potential functions in two variables

Umbilic catastrophes (“navel”) are examples of knockout rank two disasters. In optics they are u. a. important in the focal areas (for light waves that are reflected off surfaces in three dimensions). They are closely related to the geometry of almost spherical surfaces. According to Thom, the hyperbolic umbilic catastrophe models the breaking of a wave and the elliptic umbilic the emergence of hair-like structures.

Hyperbolic-umbilic disaster is

${\displaystyle V=x^{3}+y^{3}+axy+bx+cy\,}$

Elliptic umbilic disaster is

${\displaystyle V=x^{3}/3-xy^{2}+a(x^{2}+y^{2})+bx+cy\,}$

Parabolic-umbilic disaster is

${\displaystyle V=x^{2}y+y^{4}+ax^{2}+by^{2}+cx+dy\,}$

Arnold's notation

Vladimir Arnold gave the disasters the ADE classification , which is based on deep connections to Lie groups and algebras and their Dynkin diagrams .

• A ₀ - a non-singular point . ${\displaystyle V=x}$
• A ₁ - a local extremum, either a stable minimum or an unstable maximum . ${\displaystyle V=\pm x^{2}+ax}$
• A ₂ - the fold, fold
• A ₃ - the tip, cusp
• A ₄ - the swallowtail, swallowtail
• A ₅ - the butterfly, butterfly
• A _k - an infinite sequence of shapes in a variable ${\displaystyle V=x^{k+1}+\cdots }$
• D ₄ ^- - the elliptical umbilic
• D ₄ ⁺ - the hyperbolic umbilic
• D ₅ - the parabolic umbilic
• D _k - an infinite series of other umbilic forms
• E ₆ - the symbolic umbilic ${\displaystyle V=x^{3}+y^{4}+axy^{2}+bxy+cx+dy}$
• E ₇
• E ₈

Objects in the theory of singularities also correspond to the remaining simple Lie groups (in ADE, A stands for the diagrams corresponding to the special unitary groups, D for those corresponding to the orthogonal group, E for special simple Lie groups).

literature

• Wladimir Arnold: Catastrophe Theory. Springer 1998.
• Robert Gilmore: Catastrophe Theory for Scientists and Engineers. Dover, New York 1993.
• Hermann Haken : Synergetics. Springer 1982.
• Tim Poston, Ian Stewart : Catastrophe Theory and Its Applications. Dover, New York 1998, ISBN 0-486-69271-X .
• René Thom : Structural Stability and Morphogenesis: An Outline of a General Theory of Models. Addison-Wesley, Reading, MA 1989, ISBN 0-201-09419-3 .
• J. Thompson: Instabilities and Catastrophes in Science and Engineering. Wiley, New York 1982.
• Monte Davis, Alexander Woodcock: Catastrophe Theory. Dutton, New York 1978, Pelican 1980.
• EC Zeeman: Catastrophe Theory-Selected Papers 1972–1977. Addison-Wesley, Reading, MA 1977.

Web links

• Disaster theory . In: Lexicon of Physics . Spectrum of Science, 1998 ( Spektrum.de ).
• Explanation in the Lexicon of Complexity Theory, English
• Declaration by Dujardin, English
• Lectures by EC Zeeman, a. a. Catastrophe Theory, English ( Memento from July 27, 2008 in the Internet Archive )
• Michor Elementary catastrophe theory , PDF file
• Georges Szpiro: The crash of catastrophe theory. In: NZZ. December 15, 2002 . 

Individual evidence

1. ↑ John Guckenheimer The catastrophe controversy , Mathematical Intelligencer, 1978, No. 1, pp. 15-20