# Pitchfork bifurcation

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.

upercritical case

180px|right|thumb|Supercritical case: solid lines represents stable points, while dotted linesrepresents unstable one.The normal form of the supercritical pitchfork bifurcation is:$frac\left\{dx\right\}\left\{dt\right\}=rx-x^3.$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 $x = pmsqrt\left\{r\right\}$.

ubcritical case

180px|right|thumb|Subcritical case: solid lines represents stable points, while dotted linesrepresents unstable one.The normal form for the subcritical case is:$frac\left\{dx\right\}\left\{dt\right\}=rx+x^3.$In this case, for $r<0$ the equilibrium at $x=0$ is stable, and there are two unstable equilbria at $x = pmsqrt\left\{-r\right\}$. For $r>0$ the equilibrium at $x=0$ is unstable.

Formal definition

An ODE:$dot\left\{x\right\}=f\left(x,r\right),$ described by a one parameter function $f\left(x, r\right)$ with $r in Bbb\left\{R\right\}$ satisfying::$-f\left(x, r\right) = f\left(-x, r\right),,$ (f is an odd function),

:

has a pitchfork bifurcation at $\left(x, r\right) = \left(0, r_\left\{o\right\}\right)$. The form of the pitchfork is givenby 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.

* Bifurcation theory
* Bifurcation diagram

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