Pitchfork bifurcation

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For other uses, see Pitchfork (disambiguation).

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[edit]

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

The normal form of the supercritical pitchfork bifurcation is


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[edit]

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

The normal form for the subcritical case is


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[edit]



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),

\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,
\displaystyle\frac{\part f}{\part r}(0, r_{o}) = 0, &
\displaystyle\frac{\part^2 f}{\part r \part x}(0, r_{o}) \neq 0.

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})
    < 0, & \mathrm{supercritical} \\
    > 0, & \mathrm{subcritical} 


  • 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.

See also[edit]