Pitchfork Bifurcation

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

$$f(-x,\mu) = -f(x,\mu)\hfill\eqnum$$ (1)

$$\left[{\partial f\over\partial x}\right]_{\mu=0, x=0} = 1\hfill\eqnum$$ (2)

$$\left[{\partial f\over\partial x}\right]_{\mu,x} = \left[{\partial f\over\partial x}\right]_{\mu=0,x=\mu}\hfill\eqnum$$ (3)

$$\left[{\partial^2f\over\partial x\partial\mu}\right]_{0,0} > 0\hfill\eqnum$$ (4)

$$\left[{\partial^3 f\over\partial\mu^3}\right]_{\mu=0, x=0} < 0.\hfill\eqnum$$ (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

$$\dot x=\mu x-x^3$$ (6)

(Guckenheimer and Holmes 1997, p. 145).

See also Bifurcation

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.