Editing Catastrophe theory (section)

===Swallowtail catastrophe===
[[File:Swallowtail catastrophe animation gif.gif|thumb|Swallowtail catastrophe, with surface <math>z = -y^4 - 0.1x^4 + (1-x^2)y^2 + 0.4xy</math>]]
[[File:Smallow tail.jpg|thumb|right|160px|Swallowtail catastrophe surface]]
:<math>V = x^5 + ax^3 + bx^2 + cx \, </math>

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 {{nowrap|''a'' &gt; 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 {{nowrap|''a'' &lt; 0}}, therefore disappear at the swallowtail point, to be replaced with only a single surface of fold bifurcations remaining. [[Salvador Dalí|Salvador Dalí's]] last painting, ''[[The Swallow's Tail]]'', was based on this catastrophe.