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.
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 .
The normal form for the subcritical case is
In this case, for the equilibrium at is stable, and there are two unstable equilbria at . For the equilibrium at is unstable.
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:
- 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.