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Catastrophe theory

    In mathematics, catastrophe theory is a branch of bifurcation theory in the study of dynamical systems. 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 in the 1970s not least due to the efforts of Christopher Zeeman, considers the special case where the long-run stable solution can be identified with the minimum of a smooth, well-defined potential function (Lyapunov function).
    Small changes in parameters can cause previously stable equilibria to disappear, leading to a large and sudden transition 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

    Thom showed that if the potential function depends on three or fewer active variables, and five or fewer active parameters, then there are only seven geometries are generic for catastrophe bifurcations.
    The names Thom gave these are now presented, along with archetypical forms for potential functions which display them.

    Fold catastrophe

    V = x^3 + ax,

    Cusp catastrophe

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

    Swallowtail catastrophe

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

    Butterfly catastrophe

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

    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 ,

    See also

    See also: broken symmetry, tipping point, phase transition, domino effect, snowball effect, butterfly effect, spontaneous symmetry breaking, singularity theory

    References

    • Arnold, Vladimir Igorevich. Catastrophe Theory, 3rd ed. Berlin: Springer-Verlag, 1992.
    • Gilmore, Robert. Catastrophe Theory for Scientists and Engineers. New York: Dover, 1993.
    • Poston, T. and Stewart, Ian. Catastrophe: Theory and Its Applications. New York: Dover, 1998.
    • 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.
    • 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.
    • Zeeman, E.C. Catastrophe Theory-Selected Papers 1972-1977. Reading, MA: Addison-Wesley, 1977.

    External links

    Catastrophe Theory: CompLexicon [1]