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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 , there is one stable equilibrium at . For there is an unstable equilibrium at , and two stable equilibria at .

## Subcritical case

The normal form for the subcritical case is

In this case, for the equilibrium at is stable, and there are two unstable equilibria at . For the equilibrium at is unstable.

## Formal definition

An ODE

described by a one parameter function with satisfying:

- (f is an odd function),

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. For example, the negative of the first ODE above, , faces the same direction as the first picture but reverses the stability.

## See also

## References

- Steven Strogatz,
*Non-linear Dynamics and Chaos: With applications to Physics, Biology, Chemistry and Engineering*, Perseus Books, 2000. - S. Wiggins,
*Introduction to Applied Nonlinear Dynamical Systems and Chaos*, Springer-Verlag, 1990.