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Encyclopedia > Catastrophe theory

In mathematics, catastrophe theory is a branch of bifurcation theory in the study of dynamical systems; it is also a particular special case of more general singularity theory in geometry. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... In mathematics, specifically in the study of dynamical systems, a bifurcation occurs when a small smooth change made to the parameter values (the bifurcation parameters) of a system causes a sudden qualitative or topological change in the systems long-term dynamical behaviour. ... The Lorenz attractor is an example of a non-linear dynamical system. ... For non-mathematical singularity theories, see singularity. ... For other uses, see Geometry (disambiguation). ...


Bifurcation theory studies and classifies phenomena characterized by sudden shifts in behavior arising from small changes in circumstances, analysing how the qualitative nature of equation solutions depends on the parameters that appear in the equation. This may lead to sudden and dramatic changes, for example the unpredictable timing and magnitude of a landslide. Qualitative is an important qualifier in the following subject titles: Qualitative identity Qualitative marketing research Qualitative method Qualitative research THE BIG J This is a disambiguation page — a list of pages that otherwise might share the same title. ... The magnitude of a mathematical object is its size: a property by which it can be larger or smaller than other objects of the same kind; in technical terms, an ordering of the class of objects to which it belongs. ... This article is about geological phenomenon. ...


Catastrophe theory, which was originated with the work of the French mathematician René Thom in the 1960s, and became very popular due to the efforts of Christopher Zeeman in the 1970s, considers the special case where the long-run stable equilibrium can be identified with the minimum of a smooth, well-defined potential function (Lyapunov function). René Thom (September 2, 1923 - October 25, 2002) was a French mathematician and founder of the catastrophe theory. ... Sir Erik Christopher Zeeman (born February 4, 1925), is a mathematician known for work in geometric topology and singularity theory. ... It has been suggested that this article or section be merged with Potential. ... Lyapunov functions are of interest in mathematics, especially in stability theory. ...


Small changes in certain parameters of a nonlinear system can cause equilibria to appear or disappear, or to change from attracting to repelling and vice versa, leading to large and sudden changes of the behaviour of the system. However, examined in a larger parameter space, catastrophe theory reveals that such bifurcation points tend to occur as part of well-defined qualitative geometrical structures.

Contents

Elementary catastrophes

Catastrophe theory analyses degenerate critical points of the potential function — points where not just the first derivative, but one or more higher derivatives of the potential function are also zero. These are called the germs of the catastrophe geometries. The degeneracy of these critical points can be unfolded by expanding the potential function as a Taylor series in small perturbations of the parameters. In mathematics, a germ is an equivalence class of continuous functions from one topological space to another (often from the real line to itself), in which one point x0 in the domain has been singled out as privileged. ... Series expansion redirects here. ...


When the degenerate points are not merely accidental, but are structurally stable, the degenerate points exist as organising centres for particular geometric structures of lower degeneracy, with critical features in the parameter space around them. If the potential function depends on two or fewer active variables, and four or fewer active parameters, then there are only seven generic structures for these bifurcation geometries, with corresponding standard forms into which the Taylor series around the catastrophe germs can be transformed by diffeomorphism (a smooth transformation whose inverse is also smooth). These seven fundamental types are now presented, with the names that Thom gave them. In mathematics, structural stability is an aspect of stability theory concerning whether a given function is sensitive to a small perturbation. ... In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ...


Potential functions of one active variable

Fold catastrophe

Stable and unstable pair of extrema disappear at a fold bifurcation
Stable and unstable pair of extrema disappear at a fold bifurcation
V = x^3 + ax,

At negative values of a, the potential has two extrema - one stable, and one unstable. If the parameter a is slowly increased, the system can follow the stable minimum point. But at a=0 the stable and unstable extrema meet, and annihilate. This is the bifurcation point. At a>0 there is no longer a stable solution. If a physical system is followed through a fold bifurcation, one therefore finds that as a reaches 0, the stability of the a<0 solution is suddenly lost, and the system will make a sudden transition to a new, very different behaviour. This bifurcation value of the parameter a is sometimes called the tipping point.
Image File history File links Fold_bifurcation. ... The phrase tipping point or angle of repose is a sociology term that refers to that dramatic moment when something unique becomes common. ...


Cusp catastrophe

Diagram of cusp catastrophe, showing curves (brown, red) of x satisfying dV / dx = 0 for parameters (a,b), drawn for parameter b continuously varied, for several values of parameter a. Outside the cusp locus of bifurcations (blue), for each point (a,b) in parameter space there is only one extremising value of x. Inside the cusp, there are two different values of x giving local minima of V(x) for each (a,b), separated by a value of x giving a local maximum.
Diagram of cusp catastrophe, showing curves (brown, red) of x satisfying dV / dx = 0 for parameters (a,b), drawn for parameter b continuously varied, for several values of parameter a. Outside the cusp locus of bifurcations (blue), for each point (a,b) in parameter space there is only one extremising value of x. Inside the cusp, there are two different values of x giving local minima of V(x) for each (a,b), separated by a value of x giving a local maximum.
Cusp shape in parameter space (a,b) near the catastrophe point, showing the locus of fold bifurcations separating the region with two stable solutions from the region with one.
Cusp shape in parameter space (a,b) near the catastrophe point, showing the locus of fold bifurcations separating the region with two stable solutions from the region with one.
Pitchfork bifurcation at a=0 on the surface b=0
Pitchfork bifurcation at a=0 on the surface b=0

The cusp geometry is very common, when one explores what happens to a fold bifurcation if a second parameter, b, is added to the control space. Varying the parameters, one finds that there is now a curve (blue) of points in (a, b) space where stability is lost, where the stable solution will suddenly jump to an alternate outcome. Image File history File links Cusp_catastrophe. ... Image File history File links Cusp_shape. ... Image File history File links Pitchfork_bifurcation_left. ...


But in a cusp geometry the bifurcation curve loops back on itself, giving a second branch where this alternate solution itself loses stability, and will make a jump back to the original solution set. By repeatedly increasing b and then decreasing it, one can therefore observe hysteresis loops, as the system alternately follows one solution, jumps to the other, follows the other back, then jumps back to the first. A system with hysteresis can be summarised as a system that may be in any number of states, independent of the inputs to the system. ...


However, this is only possible in the region of parameter space a<0. As a is increased, the hysteresis loops become smaller and smaller, until above a=0 they disappear altogether (the cusp catastrophe), and there is only one stable solution.


One can also consider what happens if one holds b constant and varies a. In the symmetrical case b=0, one observes a pitchfork bifurcation as a is reduced, with one stable solution suddenly splitting into two stable solutions and one unstable solution as the physical system passes to a<0 through the cusp point a=0, b=0 (an example of spontaneous symmetry breaking). Away from the cusp point, there is no sudden change in a physical solution being followed: when passing through the curve of fold bifurcations, all that happens is an alternate second solution becomes available. In bifurcation theory, a field within mathematics, a pitchfork bifurcation is a special case zero-eigenvalue bifurcation. ... Spontaneous symmetry breaking in physics takes place when a system that is symmetric with respect to some symmetry group goes into a vacuum state that is not symmetric. ...


A famous suggestion is that the cusp catastrophe can be used to model the behaviour of a stressed dog, which may respond by becoming cowed or becoming angry. The suggestion is that at moderate stress (a>0), the dog will exhibit a smooth transition of response from cowed to angry, depending on how it is provoked. But higher stress levels correspond to moving to the region (a<0). Then, if the dog starts cowed, it will remain cowed as it is irritated more and more, until it reaches the 'fold' point, when it will suddenly, discontinuously snap through to angry mode. Once in 'angry' mode, it will remain angry, even if the direct irritation parameter is considerably reduced.


Another application example is for the outer sphere electron transfer frequently encountered in chemical and biological systems (Xu, F. Application of catastrophe theory to the ∆G to -∆G relationship in electron transfer reactions. Zeitschrift für Physikalische Chemie Neue Folge 166, 79-91 (1990)). Outer sphere electron transfer is an inorganic reaction mechanism defined as electron transfer (or redox) in which the coordination shells of the two metal centers remain intact during the electron transfer event. ...


Fold bifurcations and the cusp geometry are by far the most important practical consequences of catastrophe theory. They are patterns which reoccur again and again in physics, engineering and mathematical modelling.


The remaining simple catastrophe geometries are very specialised in comparison, and presented here only for curiosity value.


Swallowtail catastrophe

V = x^5 + ax^3 + bx^2 + cx ,

The control parameter space is three dimensional. The bifurcation set in parameter space is made up of three surfaces of fold bifurcations, which meet in two lines of cusp bifurcations, which in turn meet at a single swallowtail bifurcation point.


As the parameters go through the surface of fold bifurcations, one minimum and one maximum of the potential function disappear. At the cusp bifurcations, two minima and one maximum are replaced by one minimum; beyond them the fold bifurcations disappear. At the swallowtail point, two minima and two maxima all meet at a single value of x. For values of a>0, beyond the swallowtail, there is either one maximum-minimum pair, or none at all, depending on the values of b and c. Two of the surfaces of fold bifurcations, and the two lines of cusp bifurcations where they meet for a<0, therefore disappear at the swallowtail point, to be replaced with only a single surface of fold bifurcations remaining. Salvador Dalí's last painting, The Swallow's Tail, was based on this catastrophe. Salvador Domingo Felipe Jacinto Dalí i Domènech, 1st Marquis of Púbol (May 11, 1904 – January 23, 1989), was a Spanish surrealist painter of Catalan descent born in Figueres, Catalonia (Spain). ... La queue daronde - Série des catastrophes (The Swallows Tail - Series on Catastrophes) was the last painting of Salvador Dalí, done in May 1983. ...


Butterfly catastrophe

V = x^6 + ax^4 + bx^3 + cx^2 + dx ,

Depending on the parameter values, the potential function may have three, two, or one different local minima, separated by the loci of fold bifurcations. At the butterfly point, the different 3-surfaces of fold bifurcations, the 2-surfaces of cusp bifurcations, and the lines of swallowtail bifurcations all meet up and disappear, leaving a single cusp structure remaining when a>0


Potential functions of two active variables

Umbilic catastrophes are examples of corank 2 catastrophes. They can be observed in optics in the focal surfaces created by light reflecting off a surface in three dimensions and are intimately connected with the geometry of nearly spherical surfaces. Thom proposed that the Hyperbolic umbilic catastrophe modeled the breaking of a wave and the elliptical umbilic modeled the creation of hair like structures. For the book by Sir Isaac Newton, see Opticks. ...


Hyperbolic umbilic catastrophe

V = x^3 + y^3 + axy + bx + cy ,

Elliptic umbilic catastrophe

Parabolic umbilic catastrophe

V = x^2y + y^4 + ax^2 + by^2 + cx + dy ,

Arnold's notation

Vladimir Arnol'd gave the catastrophes the ADE classification, due to a deep connection with simple Lie groups. Vladimir Igorevich Arnold (Влади́мир И́горевич Арно́льд, born June 12, 1937 in Odessa, USSR) is one of the worlds most prolific mathematicians. ... In mathematics, the ADE classification is the complete list of simply laced groups or other mathematical objects satisfying analogous axioms. ... In mathematics, a simple Lie group is a Lie group which is also a simple group. ...

  • A0 - a non singular point: V = x.
  • A1 - a local extrema, either a stable minimum or unstable maximum V = pm x^2 + a x.
  • A2 - the fold
  • A3 - the cusp
  • A4 - the swallowtail
  • A5 - the butterfly
  • Ak - an infinite sequence of one variable forms V=x^{k+1}+cdots
  • D4- - the elliptical umbilic
  • D4+ - the hyperbolic umbilic
  • D5 - the parabolic umbilic
  • Dk - an infinite sequence of further umbilic forms
  • E6 - the symbolic umbilic V = x3 + y4 + axy2 + bxy + cx + dy + ey2
  • E7
  • E8

There are objects in singularity theory which correspond to most of the other simple Lie groups.


See also

Broken symmetry is a concept used in mathematics and physics when an object breaks either rotational symmetry or translational symmetry. ... The phrase tipping point or angle of repose is a sociology term that refers to that dramatic moment when something unique becomes common. ... This diagram shows the nomenclature for the different phase transitions. ... The domino effect refers to a small change which will cause a similar change nearby, which then will cause another similar change, and so on in linear sequence, by analogy to a falling row of dominoes standing on end. ... Production Order Snowball Effect is a SpongeBob SquarePants episode from season three. ... Point attractors in 2D phase space. ... Spontaneous symmetry breaking in physics takes place when a system that is symmetric with respect to some symmetry group goes into a vacuum state that is not symmetric. ... For other uses, see Chaos Theory (disambiguation). ...

References

  • Arnol'd, Vladimir Igorevich. Catastrophe Theory, 3rd ed. Berlin: Springer-Verlag, 1992.
  • Gilmore, Robert. Catastrophe Theory for Scientists and Engineers. New York: Dover, 1993.
  • Postle, Denis. Catastrophe Theory – Predict and avoid personal disasters. Fontana Paperbacks 1980. ISBN 0-00-635559-5
  • Poston, T. and Stewart, Ian. Catastrophe: Theory and Its Applications. New York: Dover, 1998. ISBN 0-486-69271-X.
  • Sanns, Werner. Catastrophe Theory with Mathematica: A Geometric Approach. Germany: DAV, 2000.
  • Saunders, Peter Timothy. An Introduction to Catastrophe Theory. Cambridge, England: Cambridge University Press, 1980.
  • Thom, René. Structural Stability and Morphogenesis: An Outline of a General Theory of Models. Reading, MA: Addison-Wesley, 1989. ISBN 0-201-09419-3.
  • Thompson, J. Michael T. Instabilities and Catastrophes in Science and Engineering. New York: Wiley, 1982.
  • Woodcock, Alexander Edward Richard and Davis, Monte. Catastrophe Theory. New York: E. P. Dutton, 1978. ISBN 0525078126.
  • Zeeman, E.C. Catastrophe Theory-Selected Papers 1972-1977. Reading, MA: Addison-Wesley, 1977.

Vladimir Igorevich Arnold (Russian: Влади́мир И́горевич Арно́льд, born June 12, 1937 in Odessa, USSR) is one of the worlds most prolific mathematicians. ... Ian Stewart, FRS (b. ... René Thom (September 2, 1923 - October 25, 2002) was a French mathematician and founder of the catastrophe theory. ... Sir Erik Christopher Zeeman (born February 4, 1925), is a mathematician known for work in geometric topology and singularity theory. ...

External links


  Results from FactBites:
 
Catastrophe Theory (134 words)
Originated by the French mathematician Rene Thom in the 1960s, catastrophe theory is a special branch of dynamical systems theory.
Catastrophes are bifurcations between different equilibria, or fixed point attractors.
Catastrophe theory has been applied to a number of different phenomena, such as the stability of ships at sea and their capsizing, bridge collapse, and, with some less convincing success,
  More results at FactBites »

 
 

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