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

For ${\displaystyle r<0}$, there is one stable equilibrium at ${\displaystyle x=0}$. For ${\displaystyle r>0}$ there is an unstable equilibrium at ${\displaystyle x=0}$, and two stable equilibria at ${\displaystyle x=\pm {\sqrt {r}}}$.

The normal form for the subcritical case is

In this case, for ${\displaystyle r<0}$ the equilibrium at ${\displaystyle x=0}$ is stable, and there are two unstable equilibria at ${\displaystyle x=\pm {\sqrt {-r}}}$. For ${\displaystyle r>0}$ the equilibrium at ${\displaystyle x=0}$ is unstable.

An ODE

described by a one parameter function ${\displaystyle f(x,r)}$ with ${\displaystyle r\in \mathbb {R} }$ satisfying: