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

 \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[edit]

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[edit]

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[edit]

  • 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]