Fold Bifurcation

Let  be a one-parameter family of  map satisfying

then there exist intervals ,  and  such that

1. If , then  has two fixed points in  with the positive one being unstable and the negative one stable, and

2. If , then  has no fixed points in .

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

(Guckenheimer and Holmes 1997, p. 145).

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.