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Transcritical Bifurcation

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

$\displaystyle f(0,\mu)$ $\textstyle =$ $\displaystyle 0$ (1)
$\displaystyle \left[{\partial f\over\partial x}\right]_{\mu=0, x=0}$ $\textstyle =$ $\displaystyle 1$ (2)
$\displaystyle \left[{\partial f\over\partial x}\right]_{\mu,x}$ $\textstyle =$ $\displaystyle \left[{\partial f\over\partial x}\right]_{\mu=0,x=\mu}$ (3)
$\displaystyle \left[{\partial^2f\over\partial x\partial\mu}\right]_{0,0}$ $\textstyle >$ $\displaystyle 0$ (4)
$\displaystyle \left[{\partial^2f\over\partial\mu^2}\right]_{\mu=0, x=0}$ $\textstyle >$ $\displaystyle 0.$ (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.
\end{displaymath} (6)

(Guckenheimer and Holmes 1997, p. 145).

See also Bifurcation


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.

© 1996-9 Eric W. Weisstein