Pitchfork bifurcation

In bifurcation theory, a field within mathematics, a pitchfork bifurcation is a particular type of local bifurcation.

Definition

For a one parameter function $f(x,r)$ with $r\in {\mathbb {R}}$ satisfying:

$f(x,r)=-f(-x,r)$ (f is an odd function),

${\frac {\partial f}{\partial x}}(0,r_{o})=0,{\frac {\partial ^{2}f}{\partial x^{2}}}(0,r_{o})=0,{\frac {\partial ^{3}f}{\partial x^{3}}}(0,r_{o})\neq 0,$

${\frac {\partial f}{\partial r}}(0,r_{o})=0,{\frac {\partial ^{2}f}{\partial r\partial x}}(0,r_{o})\neq 0.$

has a pitchfork bifucation at $(x,r)=(0,r_{o})$. The form of the pitchfork is given by the sign of the third derivative:

${\frac {\partial ^{3}f}{\partial x^{3}}}(0,r_{o})\left\{{\begin{matrix}<0,&\mathrm {supercritical} \\>0,&\mathrm {subcritical} \end{matrix}}\right.$

Supercritical

The normal form of the supercritical case is

${\frac {dx}{dt}}=rx-x^{3}.$

For negative values of $r$, there is one stable equilibrium at $x=0.$ At the bifurcation point $(x=0)$, two stable equilibriua branch off.

Subcritical

The normal form of the subcritical case is

${\frac {dx}{dt}}=rx+x^{3}.$