Editing Catastrophe theory (section)

==Elementary catastrophes==
Catastrophe theory analyzes ''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 [[germ (mathematics)|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 [[structural stability|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).{{Citation needed|date=May 2010}}  These seven fundamental types are now presented, with the names that Thom gave them.