A pitchfork bifurcation is a local bifurcation of equilibrium solutions in a one-parameter family of ordinary differential equations
 |
(1)
|
in which one equilibrium branch meets two additional branches at a critical value of the parameter 
. In a bifurcation diagram plotting the equilibrium value
of 
against 
,
the three branches have the shape of a pitchfork. In the figure above, solid curves
denote linearly stable equilibria and dashed curves denote unstable equilibria.
There is no special significance to placing the bifurcation at 
. If it occurs at 
, the coordinate shifts 
and 
move the critical point to the origin.
After relabeling 
and 
as 
and 
,
a family with reflection symmetry
satisfies
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
where subscripts denote partial derivatives. These symmetry and nondegeneracy conditions give a pitchfork bifurcation (Rasband 1990, p. 31). The signs of the last two derivatives determine its orientation and whether it is supercritical or subcritical (Guckenheimer and Holmes 1997, pp. 145 and 149-150).
The supercritical normal form is
 |
(6)
|
For 
,
the only equilibrium is 
, and it is stable. For 
, the equilibrium 
is unstable and the two new equilibria 
are stable. The subcritical normal form is
 |
(7)
|
For 
,
the central equilibrium 
is stable and the two equilibria 
are unstable, while for 
only the unstable central equilibrium remains. Reversing
the direction of the parameter presents this as one
unstable equilibrium splitting into three, with a stable middle branch and two unstable
outer branches. These stability statements follow by linear
stability analysis of 
.
An exact pitchfork is commonly enforced by symmetry; a perturbation that breaks the symmetry generally unfolds the pitchfork into a different local bifurcation pattern.
