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catastrophe theory

(kə′tas·trə·fē ′thē·ə·rē)

(mathematics) A theory of mathematical structure in which smooth continuous inputs lead to discontinuous responses.

 
 
Sci-Tech Encyclopedia: Catastrophe theory

A theory of mathematical structure in which smooth continuous inputs lead to discontinuous responses. Water suddenly boils, ice melts, a building crashes to the ground, or the earth unexpectedly buckles and quakes. The French mathematician René Thom conceived and developed an eclectic collection of ideas into catastrophe theory. His idea was to establish a new basis for a more mathematical approach to biology. Connotations of disaster are misleading, since Thorn's intention was to emphasize sudden, abrupt changes.

Advanced areas of modern mathematics, including algebraic geometry, differential topology, and dynamical system theory, contributed to the creation of catastrophe theory. A complete mathematical theory exists for the elementary catastrophes, which can be written as the gradient of an energylike function. The physical, chemical, and engineering applications are less developed, although many are known in optics, laser theory, thermodynamics, elasticity, and chemical reaction theory. The Thom classification theorem gives exactly seven elementary catastrophes. Although a theory of generalized catastrophes exists, which extends the theory beyond gradient systems, it is not nearly as well developed mathematically or physically as that of elementary catastrophes. It does include remarkable examples of chaos (or stochastic behavior) in the solutions to nonlinear deterministic equations. These solutions include strange attractors and omega explosions among the examples of nonelementary catastrophes. See also Geometry; Period doubling; Topology.

There are two important aspects of catastrophe theory which are frequently overlooked or misconstrued. One is that as a rigorous mathematical theory the characteristic catastrophe features can be proved. These features include: jumps in the response; hysteresis or a path dependence in the response, representing a storage of energy for some paths; divergence, where a small path change produces a large response change (as if a source or sink were crossed); and type changes in the response, where a smooth response occurs on one path which becomes discontinuous along a nearby path.

All of these features are topological, so that they are independent of the coordinates used to describe the potential. They are, therefore, qualitative features of the solutions. Some critics have concluded that because these aspects were qualitative, they could not be quantitative. This is contradicted by the solid and growing body of quantitative studies in catastrophe theory. (Problems in quantum optics, thermodynamics, and scattering theory have all been clarified by catastrophe theory.)

 
Food and Fitness: catastrophe theory

A mathematical model developed by the French mathematician, René Thom, to show how the interaction of varying factors produce sudden, dramatic changes. Sports psychologists use catastrophe theory to explain why athletes subjected to a critical level of stress experience a huge and sudden loss of performance.

Among athletes, two main factors are associated with stress: physiological arousal (changes in heart rate, sweating, adrenaline secretion etc.) and cognitive anxiety (i.e. mental anxiety). The relationship between these two factors and the performance of an athlete can be depicted in a 3-D graph, with the surface shape representing performance (figure 21). The graph shows that when stress increases up to a critical level, performance improves. At the critical level, the surface folds so that more than one level of performance can occur; an athlete's performance can leap unexpectedly from one level to the other. Beyond the critical level, further increases in stress result in poorer performances.

Figure 21
Figure 21

 
Encyclopedia of Public Health: Catastrophe Theory

Catastrophe theory is the mathematical theory that explains the observation that small incremental changes in the value of a variable in a natural system can lead to sudden large changes in the state of the system. The best-known, everyday example is the change in the state of the chemical H2O from solid (ice) to liquid (water) to gas (steam). The same processes occur in nature with many other chemical substances. In biology, medical practice, and public health there are many examples of catastrophe theory in operation. They include certain stages in the process of carcinogenesis and spread of cancer, in gene frequencies in populations, and in phases in the development, continuation, and decline and disappearance of epidemics. The same processes operate in the dissemination of ideas, innovations, and fashions.

The word "catastrophe," with its suggestion that the outcome is always undesirable, may have been an unhappy choice to describe this process. While this is certainly the case in the explosive onset of many epidemics, the same mathematical process operates in reverse when an epidemic or epidemic disease virtually disappears quite suddenly from a population. This happens when the balance of susceptible and immune individuals shifts from the proportion required to sustain an epidemic to a marginally smaller proportion where the probability of transmission of an infectious agent to a susceptible host falls below the critical level required to sustain the epidemic. Catastrophe theory should not be confused with chaos theory, although both may operate together in some circumstances.

(SEE ALSO: Chaos Theory; Epidemic Theory: Herd Immunity)

— JOHN M. LAST


 
Political Dictionary: catastrophe theory

Catastrophe theory provides a systematic classification of sudden changes from one stable condition to another, applicable to phenomena as disparate as the freezing of a liquid, the collapse of an empire, or the buckling of metal or a prison riot. Developed by 1965, the theory began to be tentatively applied to the social sciences by Christopher Zeeman and others in the following decade, and became an object of popular controversy after 1975. Its appeal to non-mathematicians was twofold. First, the mathematics of surfaces, topology, is more a qualitative than a quantitative field, yielding ideas of great generality which non-mathematicians are able to grasp through spatial intuition. Secondly, catastrophe theory offered an explanation of just those kinds of discontinuous change and radical divergence from nearly identical initial conditions that had seemed most resistant to scientific explanation in the Newtonian tradition and were thought peculiarly characteristic of social and political phenomena. Like chaos theory a decade later, catastrophe theory has intrigued students of politics without achieving an assured place in the discipline, having had more success as a heuristic device than in detailed applications. Its impact has accordingly been less than that of game theory.

— Charles Jones

 
Britannica Concise Encyclopedia: catastrophe theory

Branch of mathematics (considered a branch of geometry) that explores how gradual changes to a system produce sudden, drastic results (though usually not as dire as the name suggests). A simple example is how a plastic coffee stirrer subjected to gradually increasing pressure from both ends will suddenly buckle in one direction or another. Other "catastrophes" include optical phenomena such as reflection or refraction of light through moving water. More speculatively, ideas from catastrophe theory have been applied by social scientists to such situations as the sudden eruption of mob violence.

For more information on catastrophe theory, visit Britannica.com.

 
Philosophy Dictionary: catastrophe theory

Mathematical theory pioneered by the French mathematician René Thom, treating abrupt changes or discontinuities.

 
Archaeology Dictionary: catastrophe theory

[Th]

A mathematical theory developed by René Thom that is concerned with modelling non-linear interactions within systems that can produce sudden and dramatic effects from apparently small changes in one variable. It is argued that there is only a limited number of ways in which such changes can take place. Colin Renfrew has taken what may be referred to as elementary catastrophes as models with which to explore major changes in the archaeological record, for example the collapse of Mycenaean Greece and the end of the Roman empire.
 
Sports Science and Medicine: catastrophe theory

A theory based on a mathematical model developed by the French mathematician René Thom to understand change and discontinuity in systems. It has been used to explain how, through the interaction of various factors, a small change in one of the factors affecting a system can lead to a catastrophic change in the system. Sport psychologists have applied the theory to the development of stress in athletes during competition. Such stress results from a complex interaction between physiological arousal (as reflected by changes in heart rate, sweating, and adrenaline secretion) and cognitive anxiety (i.e. mental anxiety). When athletes are subjected to a small increase in stress above a critical level, they may experience a huge and sudden loss of performance. Compare inverted U hypothesis.

Catastrophe theory
Catastrophe theory

 
Veterinary Dictionary: catastrophe theory

The mathematical basis for the study of large changes in a total system which may result from small changes in a critical variable in the system.

 
Wikipedia: catastrophe theory
This article refers to the study of how the behaviour of dynamical systems can change drastically with variations in certain parameters. For other meanings of the word catastrophe, including catastrophe modeling in insurance, see catastrophe (disambiguation).

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.

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.

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).

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.

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.

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 three or fewer active variables, and five 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.

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.

Cusp catastrophe

V = x^4 + ax^2 + bx \,
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.

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.

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.

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)).

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.

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.

Hyperbolic umbilic catastrophe

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

Elliptic umbilic catastrophe

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

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.

  • A0 - a non singular point: V = x.
  • A1 - a local extrema, either a stable minimum or unstable maximum V = ±x2 + ax.
  • 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
  • E7
  • E8

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

See also

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.

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