The butterfly catastrophe curve is the plane curve with parametric equations
 |  |  |
(1)
|
 |  |  |
(2)
|
where 
and 
are real numbers with 
and 
ranges over the real numbers
(von Seggern 1993, p. 94). The smallest algebraic
curve containing this parametrized locus has degree
5. Its implicit equation therefore defines a quintic
curve; its real points can also include isolated points not traced by real 
.
It is a two-control-parameter section of the butterfly catastrophe. For fixed 
and 
,
consider the potential 
.
The curve consists of the parameter pairs 
for which 
has a degenerate critical
point, meaning that 
for some real 
. Solving these two conditions for 
and 
gives the displayed parametric
equations. The curve is therefore a planar discriminant section, rather than
the full four-control-parameter butterfly catastrophe.
