In bifurcation theory, a field within mathematics, a pitchfork bifurcation is a particular type of local bifurcation where the system transitions from one fixed point to three fixed points. Pitchfork bifurcations, like Hopf bifurcations, have two types – supercritical and subcritical.
In continuous dynamical systems described by ODEs - i.e. flows - pitchfork bifurcations occur generically in systems with symmetry.
The normal form of the supercritical pitchfork bifurcation is
| dx | |
| dt |
=rx-x3.
r<0
x=0
r>0
x=0
x=\pm\sqrt{r}
The normal form for the subcritical case is
| dx | |
| dt |
=rx+x3.
r<0
x=0
x=\pm\sqrt{-r}
r>0
x=0
An ODE
| x |
=f(x,r)
f(x,r)
r\inR
-f(x,r)=f(-x,r)
\begin{align}
| \partialf | |
| \partialx |
(0,r0)&=0,&
| \partial2f | |
| \partialx2 |
(0,r0)&=0,&
| \partial3f | |
| \partialx3 |
(0,r0)& ≠ 0,\\[5pt]
| \partialf | |
| \partialr |
(0,r0)&=0,&
| \partial2f | |
| \partialx\partialr |
(0,r0)& ≠ 0. \end{align}
has a pitchfork bifurcation at
(x,r)=(0,r0)
| \partial3f | |
| \partialx3 |
(0,r0)\begin{cases} <0,&supercritical\\ >0,&subcritical\end{cases}
Note that subcritical and supercritical describe the stability of the outer lines of the pitchfork (dashed or solid, respectively) and are not dependent on which direction the pitchfork faces. For example, the negative of the first ODE above,
| x |
=x3-rx