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 two or fewer active variables, and four (resp. five) or fewer active parameters, then there are only seven (resp. eleven) 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.
Read more about this topic: Catastrophe Theory
Famous quotes containing the words elementary and/or catastrophes:
“As if paralyzed by the national fear of ideas, the democratic distrust of whatever strikes beneath the prevailing platitudes, it evades all resolute and honest dealing with what, after all, must be every healthy literature’s elementary materials.”
—H.L. (Henry Lewis)
“Science fiction writers foresee the inevitable, and although problems and catastrophes may be inevitable, solutions are not.”
—Isaac Asimov (1920–1992)