Pitchfork bifurcation
From Wikipedia, the free encyclopedia
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
The normal form of the supercritical pitchfork bifurcation is
- <math> \frac{dx}{dt}=rx-x^3. </math>
For negative values of <math>r</math>, there is one stable equilibrium at <math>x = 0</math>. For <math>r>0</math> there is an unstable equilibrium at <math>x = 0</math>, and two stable equilibria at <math>x = \pm\sqrt{r}</math>.
[edit] Subcritical case
The normal form for the subcritical case is
- <math> \frac{dx}{dt}=rx+x^3. </math>
In this case, for <math>r<0</math> the equilibrium at <math>x=0</math> is stable, and there are two unstable equilbria at <math>x = \pm\sqrt{-r}</math>. For <math>r>0</math> the equilibrium at <math>x=0</math> is unstable.
[edit] Formal definition
An ODE
- <math> \dot{x}=f(x,r)</math>
described by a one parameter function <math>f(x, r)</math> with <math> r \in \Bbb{R}</math> satisfying:
- <math> f(x, r) = -f(-x, r) </math> (f is an odd function),
- <math> \frac{\part f}{\part x}(0, r_{o}) = 0 ,
\frac{\part^2 f}{\part x^2}(0, r_{o}) = 0, \frac{\part^3 f}{\part x^3}(0, r_{o}) \neq 0, </math>
- <math> \frac{\part f}{\part r}(0, r_{o}) = 0,
\frac{\part^2 f}{\part r \part x}(0, r_{o}) \neq 0.</math>
has a pitchfork bifucation at <math>(x, r) = (0, r_{o})</math>. The form of the pitchfork is given by the sign of the third derivative:
- <math> \frac{\part^3 f}{\part x^3}(0, r_{o})
\left\{
\begin{matrix}
< 0, & \mathrm{supercritical} \\
> 0, & \mathrm{subcritical}
\end{matrix}
\right. </math>
