Pitchfork bifurcation

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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 flows, that is, continuous dynamical systems described by ODEs, pitchfork bifurcations occur generically in systems with symmetry.

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[edit] Supercritical case

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

[edit] Subcritical case

Subcritical case: solid lines represents stable points, while dotted lines represents unstable one.
Subcritical case: solid lines represents stable points, while dotted lines represents unstable one.

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.

[edit] 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,ro). 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.\,\,

[edit] 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.

[edit] See also