Catastrophe theory

The mathematical catastrophe theory deals with discontinuous, abrupt changes of smooth dynamical systems. This can, even if they are looking for steady state under certain conditions, experienced abrupt, discontinuous discontinuous changes of the solution to changes in the parameters.

The catastrophe theory examines the branching behavior of these solutions ( bifurcations ) with variation of the parameters and is thus an important basis for the mathematical treatment of the theory of chaos. It is sometimes preferred to speak in the mathematics of the theory of singularities of differentiable maps, and the lurid Name avoided catastrophe theory. Main result is the classification of these singularities in seven " normal types ".

The catastrophe theory is based fundamentally on the differential geometry (or differential topology). It was developed in the late 1960s by René Thom, Vladimir Arnold, and others. It is used and enhancements, among other things in modern physics and economics, but also in linguistics and psychology, and was therefore in these areas in the 1970s, popular as a directly applicable qualitative mathematical method. Particularly active was there the English mathematician Erik Christopher Zeeman, who used the theory of ship stability to the theory of evolution. Thom even sought, as the title of his book from 1972 shows, especially applications in biology (especially of the embryo development, morphogenesis ).

• 3.1 Hyperbolic- umbilic catastrophe
• 3.2 Elliptic umbilic catastrophe
• 3.3 parabolic umbilic catastrophe

Elementary disasters

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

Not let the critical points by small fault rectification, they are called structurally stable. Their geometric structure can be classified geometries at three or fewer variables of the potential function and five or fewer parameters for this function with only seven types of ( bifurcation ). They correspond to the normal forms to which the Taylor expansion can be attributed to catastrophe germs with the help of diffeomorphisms ( differentiable maps ).

Potential functions of a variable

In practice, the Faltungsbifurkationen and the peak disaster ( cusp geometry) are the most important by far cases of catastrophe theory and occur in many cases. The remaining disasters are very specialized and are mentioned here for completeness.

Convolutional disaster ( fold catastrophe )

For negative values ​​of a, the potential function has a stable and an unstable extremum. If the parameter a is too slow, the system may follow the stable minimum. When a = 0 meet the stable and unstable extrema and lift up ( bifurcation point ). For a> 0, there is no stable solution more. A physical system at a = 0 would lose its existing for negative a stability suddenly and overturn his behavior.

Lace - disaster ( cusp catastrophe )

The cusp catastrophe occurs rather frequently when considering the behavior of convolution disaster when a second parameter b is added to the parameter space. If you change the parameters of now, there is a curve ( blue in the figure ) of points in the parameter space (a, b ) above which lost stability. Instead of an extremum there are now two of which can jump the system. If you change periodically b, one can generate " jumping back and forth " so also in the spatial domain. S is only possible for the range a < 0, the more get a zero, the smaller the hysteresis curves and eventually disappear at a = 0 throughout.

If you hold vice versa b constant and varies a is observed in the symmetric case b = 0 is a tuning fork bifurcation ( pitchfork bifurcation ) takes a from, a stable solution splits suddenly into two stable and one unstable solution when the system the cusp point a = 0, b = 0 happens to negative values ​​of a. This is an example of spontaneous symmetry breaking. Further away from the cusp point missing this sudden change in the structure of the solution and it appeared only a second possible solution.

A well-known example modeled with the cusp behavior of a stressed dog between submissiveness and aggressiveness. Under moderate stress ( a> 0) indicates the dog depending on provocation (parameter b ) a continuous transition behavior between the two behaviors. At higher stress ( region a < 0) is the dog even when attenuated provocation in eingeschüchtertem state, then suddenly toppling over when reaching the folding point in aggressive behavior, which he maintains even when reducing the provocation parameter.

Another example is the critical point (a = b = 0), taking place a crossing magnetic system ( more precisely, of a ferromagnetic system, such as iron) at below the critical temperature Tc of the non-magnetic in the ferromagnetic state. The parameter a is proportional to the temperature difference between Tc and T B is proportional to the magnetic field. Furthermore, one can very well explain the term Spontaneous symmetry breaking in this example, since the ferromagnetic state - depending on the sign of a very weak symmetry-breaking magnetic field - one of the two marked directions is preferred.

Catastrophe theory neglect the fluctuations occurring here in the vicinity of the critical point, such as the Magnetisierungsfluktionen ( cf. monodromy ).

Shape of the cusp in the parameter space (a, b ) close to a point of catastrophic, which separates the regions of one and two extremes, for a specific choice of a curve (x2 ), and B (X2).

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

Swallowtail catastrophe ( swallowtail catastrophe )

Here, the space of control parameters is three dimensional. The Bifurkationsmenge consists of three surfaces of fold catastrophes, which meet in two cusp bifurcations. This in turn meet in a single swallowtail bifurcation point.

Pass the parameters through the areas of the convolution bifurcations, a minimum and a maximum of the potential function disappears. On the cusp bifurcations two minima and a maximum to be replaced by a minimum, behind them the folding bifurcation disappears. The dovetail point, two minima and two maxima meet at a single point x. For values ​​of a> 0, the other side of the dovetail, it is dependent on the parameter values ​​of b and c is either a maximum-minimum pair or none at all. Two of the surfaces of the folding bifurcations and the two curves of cusp bifurcations disappear in the dovetail point and remains only a single area of ​​convolutional bifurcations. Salvador Dalí's last painting The dovetail based on this disaster.

Butterfly disaster ( butterfly catastrophe )

Depending on the parameters, the potential function 3, have 2 or 1 local minimum. The different areas are separated by Faltungsbifurkationen in the parameter space. At the Butterfly point to meet the various 3- surfaces of Faltungsbifurkationen, 2- surfaces of cusp bifurcations and curves of Butterfly bifurcations and disappear to left to allow only a single cusp structure for a> 0.

Potential functions in two variables

Umbilic catastrophes ( " navel " ) are examples of disasters from the co- rank two. In optics they are, inter alia, in the focus areas important (for the light waves, which are reflected at surfaces in three dimensions ). They are closely connected with the geometry of almost - spherical surfaces. After Thom Hyperbolic- umbilic catastrophe modeled the breaking of a wave and the elliptic umbilic the emergence of hair -like structures.

Hyperbolic- umbilic catastrophe

Elliptic umbilic catastrophe

Parabolic umbilic catastrophe

Arnold's notation

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

• A0 - is a non- singular point.
• A1 - a local extremum, either a stable or an unstable minimum maximum.
• A2 - the folding, fold
• A3 - the tip, cusp
• A4 - the swallowtail, swallowtail
• A5 - the butterfly, butterfly
• Ak - an infinite sequence of shapes in a Variable
• D4 - the elliptical umbilic
• D4 - the hyperbolic umbilic
• D5 - the parabolic umbilic
• Dk - an infinite sequence of further umbilic forms
• E6 - the symbolic umbilic
• E7
• E8

Also, the remaining simple Lie groups correspond to objects in the theory of singularities ( in ADE, A is the corresponding special unitary groups diagrams D for the corresponding orthogonal group E for special simple Lie groups ).