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Transcritical Bifurcation

Let f:R×R->R be a one-parameter family of C^2 maps satisfying

f(0,mu)=0
(1)
[(partialf)/(partialx)]_(mu=0,x=0)=0
(2)
[(partial^2f)/(partialxpartialmu)]_(0,0)>0
(3)
[(partial^2f)/(partialx^2)]_(mu=0,x=0)>0.
(4)

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

x^.=mux-x^2
(5)

(Guckenheimer and Holmes 1997, p. 145).

See also

Bifurcation, Pitchfork Bifurcation

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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, p. 30, 1990.

Referenced on Wolfram|Alpha

Transcritical Bifurcation

Cite this as:

Weisstein, Eric W. "Transcritical Bifurcation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TranscriticalBifurcation.html

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