Transcritical Bifurcation

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

$$f(0,\mu) = 0$$ (1)

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

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

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

$$\left[{\partial^2f\over\partial\mu^2}\right]_{\mu=0, x=0} > 0.$$ (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

$$\dot x=\mu x-x^2.$$ (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, pp. 27-28, 1990.