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{{Other uses|Pitchfork (disambiguation)}} | |||
In [[bifurcation theory]], a field within [[mathematics]], a '''pitchfork bifurcation''' is a particular type of local bifurcation. Pitchfork bifurcations, like [[Hopf bifurcation]]s have two types - supercritical or subcritical. | |||
In continuous dynamical systems described by [[Ordinary differential equation|ODEs]]—i.e. flows—pitchfork bifurcations occur generically in systems with [[symmetry in mathematics|symmetry]]. | |||
==Supercritical case== | |||
[[Image:Pitchfork bifurcation supercritical.svg|180px|right|thumb|Supercritical case: solid lines represent stable points, while dotted line | |||
represents unstable one.]] | |||
The [[normal form (bifurcation theory)|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>. | |||
==Subcritical case== | |||
[[Image:Pitchfork bifurcation subcritical.svg|180px|right|thumb|Subcritical case: solid line represents stable point, while dotted lines | |||
represent unstable ones.]] | |||
The [[normal form (bifurcation theory)|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. | |||
==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><!-- the \,\, is to force no-hinting in the antialiasing mode of pngtex - please do not remove it! --> (f is an [[odd function]]), | |||
:<math> | |||
\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} | |||
</math> | |||
has a '''pitchfork bifurcation''' 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><!-- the \,\, is to force no-hinting in the antialiasing mode of pngtex - please do not remove it! --> | |||
==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. | |||
== See also == | |||
* [[Bifurcation theory]] | |||
* [[Bifurcation diagram]] | |||
[[Category:Bifurcation theory]] |
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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 continuous dynamical systems described by ODEs—i.e. flows—pitchfork bifurcations occur generically in systems with symmetry.
Supercritical case
![](https://upload.wikimedia.org/wikipedia/commons/thumb/2/22/Pitchfork_bifurcation_supercritical.svg/180px-Pitchfork_bifurcation_supercritical.svg.png)
The normal form of the supercritical pitchfork bifurcation is
For negative values of , there is one stable equilibrium at . For there is an unstable equilibrium at , and two stable equilibria at .
Subcritical case
![](https://upload.wikimedia.org/wikipedia/commons/thumb/b/be/Pitchfork_bifurcation_subcritical.svg/180px-Pitchfork_bifurcation_subcritical.svg.png)
The normal form for the subcritical case is
In this case, for the equilibrium at is stable, and there are two unstable equilbria at . For the equilibrium at is unstable.
Formal definition
An ODE
described by a one parameter function with satisfying:
- (f is an odd function),
has a pitchfork bifurcation at . The form of the pitchfork is given by the sign of the third derivative:
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.