Pitchfork bifurcation

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

The normal form of the supercritical pitchfork bifurcation is

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

For ${\displaystyle r<0}$, there is one stable equilibrium at ${\displaystyle x=0}$. For ${\displaystyle r>0}$ there is an unstable equilibrium at ${\displaystyle x=0}$, and two stable equilibria at ${\displaystyle x=\pm {\sqrt {r}}}$.

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

The normal form for the subcritical case is

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

In this case, for ${\displaystyle r<0}$ the equilibrium at ${\displaystyle x=0}$ is stable, and there are two unstable equilibria at ${\displaystyle x=\pm {\sqrt {-r}}}$. For ${\displaystyle r>0}$ the equilibrium at ${\displaystyle x=0}$ is unstable.

An ODE

${\displaystyle {\dot {x}}=f(x,r)\,}$

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

${\displaystyle -f(x,r)=f(-x,r)\,\,}$  (f is an odd function),

${\displaystyle {\begin{aligned}{\frac {\partial f}{\partial x}}(0,r_{0})&=0,&{\frac {\partial ^{2}f}{\partial x^{2}}}(0,r_{0})&=0,&{\frac {\partial ^{3}f}{\partial x^{3}}}(0,r_{0})&\neq 0,\\[5pt]{\frac {\partial f}{\partial r}}(0,r_{0})&=0,&{\frac {\partial ^{2}f}{\partial r\partial x}}(0,r_{0})&\neq 0.\end{aligned}}}$

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

${\displaystyle {\frac {\partial ^{3}f}{\partial x^{3}}}(0,r_{0}){\begin{cases}<0,&{\text{supercritical}}\\>0,&{\text{subcritical}}\end{cases}}\,\,}$

Note that subcritical and supercritical describe the stability of the outer lines 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, ${\displaystyle {\dot {x}}=x^{3}-rx}$, faces the same direction as the first picture but reverses the stability.

• Bifurcation theory
• Bifurcation diagram

• 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.