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Pitchfork Bifurcation

Let $f:\Bbb{R}\times\Bbb{R}\to\Bbb{R}$ be a one-parameter family of $C^3$ map satisfying

$\displaystyle f(-x,\mu)$ $\textstyle =$ $\displaystyle -f(x,\mu)\hfill\eqnum$ (1)
$\displaystyle \left[{\partial f\over\partial x}\right]_{\mu=0, x=0}$ $\textstyle =$ $\displaystyle 1\hfill\eqnum$ (2)
$\displaystyle \left[{\partial f\over\partial x}\right]_{\mu,x}$ $\textstyle =$ $\displaystyle \left[{\partial f\over\partial x}\right]_{\mu=0,x=\mu}\hfill\eqnum$ (3)
$\displaystyle \left[{\partial^2f\over\partial x\partial\mu}\right]_{0,0}$ $\textstyle >$ $\displaystyle 0\hfill\eqnum$ (4)
$\displaystyle \left[{\partial^3 f\over\partial\mu^3}\right]_{\mu=0, x=0}$ $\textstyle <$ $\displaystyle 0.\hfill\eqnum$ (5)

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 Bifurcation is called a pitchfork bifurcation. An example of an equation displaying a pitchfork bifurcation is
\dot x=\mu x-x^3
\end{displaymath} (6)

(Guckenheimer and Holmes 1997, p. 145).

See also Bifurcation


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.

© 1996-9 Eric W. Weisstein