Let
be a one-parameter family of maps satisfying

(1) | |||

(2) | |||

(3) | |||

(4) | |||

(5) |

Then there are two branches, one stable and one unstable. This Bifurcation is called a transcritical bifurcation. An example of an equation displaying a transcritical bifurcation is

(6) |

**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, pp. 27-28, 1990.

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1999-05-26