Pitchfork bifurcation

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.
Supercritical case.
The normal form of the supercritical pitchfork bifurcation is
For formula_2, there is one stable equilibrium at formula_3. For formula_4 there is an unstable equilibrium at formula_3, and two stable equilibria at formula_6.
Subcritical case.
The normal form for the subcritical case is
In this case, for formula_2 the equilibrium at formula_9 is stable, and there are two unstable equilibria at formula_10. For formula_4 the equilibrium at formula_9 is unstable.
Formal definition.
An ODE
described by a one parameter function formula_14 with formula_15 satisfying:
has a pitchfork bifurcation at formula_18. The form of the pitchfork is given
by the sign of the third derivative:
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, formula_20, faces the same direction as the first picture but reverses the stability.