Pitchfork bifurcation

Bifurcation diagram of a supercritical pitchfork bifurcation. Stable fixed points are shown in solid lines, unstable points in dotted lines.

Bifurcation diagram of a subcritical pitchfork bifurcation. Stable fixed points are shown in solid lines, unstable points in dotted lines.

The pitchfork bifurcation , also called pitchfork or tuning fork bifurcation , is a specific type of bifurcation in a nonlinear system .

The normal form of the pitchfork bifurcation is:

${\displaystyle {\frac {dx}{dt}}=r\cdot x+\alpha \cdot x^{3}}$

With

${\displaystyle \alpha =\pm 1}$

where is the parameter to be varied for the occurrence of the bifurcation. ${\displaystyle r}$

For α = 1 one obtains the subcritical pitch fork bifurcation

The supercritical pitch fork bifurcation is obtained for α = −1

The pitchfork bifurcation has the following equilibrium points:

${\displaystyle {x_{1}}^{*}=0}$

${\displaystyle {x_{2}}^{*}=\pm {\sqrt {-{\frac {r}{\alpha }}}}}$

See also

• Transcritical bifurcation
• Hopf bifurcation
• Saddle node bifurcation