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

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

The normal form of the supercritical pitchfork bifurcation is

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

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

Subcritical case

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

The normal form for the subcritical case is

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

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

Formal definition

An ODE

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

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

$-f(x, r) = f(-x, r)\,\,$  (f is an odd function),
$\begin{array}{lll} \displaystyle\frac{\part f}{\part x}(0, r_{o}) = 0 , & \displaystyle\frac{\part^2 f}{\part x^2}(0, r_{o}) = 0, & \displaystyle\frac{\part^3 f}{\part x^3}(0, r_{o}) \neq 0, \\[12pt] \displaystyle\frac{\part f}{\part r}(0, r_{o}) = 0, & \displaystyle\frac{\part^2 f}{\part r \part x}(0, r_{o}) \neq 0. \end{array}$

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

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

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.