Pitchfork bifurcation

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For other uses, see Pitchfork (disambiguation).

In bifurcation theory, a field within mathematics, a pitchfork bifurcation is a particular type of local bifurcation. Pitchfork bifurcations, like Hopf bifurcations have two types - supercritical or subcritical.

In continuous dynamical systems described by ODEs—i.e. flows—pitchfork bifurcations occur generically in systems with symmetry.

Supercritical case[edit]

Supercritical case: solid lines represent stable points, while dotted line represents unstable one.

The normal form of the supercritical pitchfork bifurcation is

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

Subcritical case[edit]

Subcritical case: solid line represents stable point, while dotted lines represent unstable ones.

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[edit]


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 tines 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.


  • 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.

See also[edit]