be a one-parameter family of maps satisfying
Here, it turns out that condition (1) can be relaxed slightly, and the left-hand side of (2) has been corrected from the value
of 1 given by Rasband (1990, p. 30).
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
(Guckenheimer and Holmes 1997, p. 145).
Explore with Wolfram|Alpha
ReferencesGuckenheimer, 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.
Dynamics of Nonlinear Systems. New York: Wiley, p. 30, 1990.
on Wolfram|AlphaTranscritical Bifurcation
Cite this as:
Weisstein, Eric W. "Transcritical Bifurcation."
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TranscriticalBifurcation.html