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Date: 17-11-2021
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Let be a one-parameter family of maps satisfying
(1) |
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(2) |
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(3) |
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(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
(5) |
(Guckenheimer and Holmes 1997, p. 145).
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.
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