Dangerous Intersection

Posted on: October 13th, 2012 by jdelgre

Catastrophe theory is a subbranch of an area of mathematics called bifurcation theory, which itself is a subdiscipline of dynamical systems theory.  Catastrophe theory was founded by the famous French mathematician Rene Thom (1923 – 2002) in the late 1960’s, and became very popular in the 1970’s.  Catastrophes are essentially bifurcations (or splits) between points of equilibria, called fixed-point attractors.  Bifurcation theory studies and classifies phenomena characterized by sudden shifts in behavior arising from small changes in circumstances, analyzing how the nature of the solutions of equations depends on the parameters that appear in the equation. This may lead to dramatic changes of state, for example the unpredictable timing and magnitude of a landslide.  Some people tried to use catastrophe theory, somewhat dubiously, to explain why prisoners suddenly riot or why the mere flapping of a butterfly’s wings could precipitate the formation of a hurricane in the Caribbean ocean.

Catastrophes can be classified into various categories:  fold, cusp, butterfly, and swallowtail catastrophes for one control variable, and hyperbolic, elliptic, and parabolic catastrophes for two control variables.  Researches have tried to classify catastrophes for any number of control variables, but no classification exists for more than five control variables.  It turns out that there is a deep connection between catastrophe theory and simple lie group theory, and a classification was worked out by the mathematician Vladimir Arnold based on simply laced Dynkin diagrams, the so-called ADE classification.

The article ‘Dangerous Intersection’ which appeared in the New York Times on October 8, 2012 was related to me by Dr. Ann Hupert.  Read more…

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