Fold Bifurcation

Let $f:\Bbb{R}\times\Bbb{R}\to\Bbb{R}$ be a one-parameter family of $C^2$ Map satisfying

$$\begin{eqnarray*} f(0,0)&=&0\\ \left[{\partial f\over\partial x}\right]_{\mu=... ...\\ \left[{\partial f\over\partial\mu}\right]_{\mu=0, x=0}&>&0, \end{eqnarray*}$$

then there exist intervals $(\mu_1,0)$, $(0,\mu_2)$ and $\epsilon>0$ such that

1. If $\mu\in(\mu_1,0)$, then $f_\mu(x)$ has two fixed points in $(-\epsilon,\epsilon)$ with the positive one being unstable and the negative one stable, and

2. If $\mu\in(0,\mu_2)$, then $f_\mu(x)$ has no fixed points in $(-\epsilon,\epsilon)$.

This type of Bifurcation is known as a fold bifurcation, sometimes also called a Saddle-Node Bifurcation or Tangent Bifurcation. An example of an equation displaying a fold bifurcation is

$$\dot x=\mu-x^2$$

(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. 145-149, 1997.

Rasband, S. N. Chaotic Dynamics of Nonlinear Systems. New York: Wiley, pp. 27-28, 1990.