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.

The normal form of the supercritical pitchfork bifurcation is For

, there is one stable equilibrium at

The normal form for the subcritical case is In this case, for

is stable, and there are two unstable equilibria at

An ODE described by a one parameter function

satisfying: has a pitchfork bifurcation at

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.

, faces the same direction as the first picture but reverses the stability.