Pitchfork bifurcation
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
![](../I/m/Pitchfork_bifurcation_supercritical.svg.png)
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
![](../I/m/Pitchfork_bifurcation_subcritical.svg.png)
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:
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.