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

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

ubcritical case

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

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),

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has a pitchfork bifurcation at $(x,\; r)\; =\; (0,\; r\_\{o\})$. The form of the pitchfork is givenby the sign of the third derivative:

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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