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Pitchfork Bifurcation

Let f:R×R->R be a one-parameter family of C^3 maps satisfying

f(-x,mu)=-f(x,mu)
(1)
(partialf)/(partialx)|_(mu=0, x=0)=0
(2)
(partial^2f)/(partialxpartialmu)|_(mu=0, x=0)>0
(3)
(partial^3f)/(partialx^3)|_(mu=0, x=0)<0
(4)

(Rasband 1990, p. 31), although condition (1) can actually be relaxed slightly. Then there are intervals having a single stable fixed point and three fixed points (two of which are stable and one of which is unstable). This type of bifurcation is called a pitchfork bifurcation.

An example of an equation displaying a pitchfork bifurcation is

x^.=mux-x^3
(5)

(Guckenheimer and Holmes 1997, p. 145).

See also

Bifurcation, Transcritical Bifurcation

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References

Guckenheimer, J. and Holmes, P. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, 3rd ed. New York: Springer-Verlag, pp. 145 and 149-150, 1997.Rasband, S. N. Chaotic Dynamics of Nonlinear Systems. New York: Wiley, p. 31, 1990.

Referenced on Wolfram|Alpha

Pitchfork Bifurcation

Cite this as:

Weisstein, Eric W. "Pitchfork Bifurcation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PitchforkBifurcation.html

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