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Bifurcation

المؤلف:  Guckenheimer, J. and Holmes, P. "Local Bifurcations.

المصدر:  Ch. 3 in Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, 2nd pr., rev. corr. New York: Springer-Verlag

الجزء والصفحة:  ...

29-8-2021

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Bifurcation

In a dynamical system, a bifurcation is a period doubling, quadrupling, etc., that accompanies the onset of chaos. It represents the sudden appearance of a qualitatively different solution for a nonlinear system as some parameter is varied. The illustration above shows bifurcations (occurring at the location of the blue lines) of the logistic map as the parameter  is varied. Bifurcations come in four basic varieties: flip bifurcation, fold bifurcation, pitchfork bifurcation, and transcritical bifurcation (Rasband 1990).

More generally, a bifurcation is a separation of a structure into two branches or parts. For example, in the plot above, the function , where  denotes the real part, exhibits a bifurcation along the negative real axis  and .

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REFERENCES:

Guckenheimer, J. and Holmes, P. "Local Bifurcations." Ch. 3 in Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, 2nd pr., rev. corr. New York: Springer-Verlag, pp. 117-165, 1983.

Lichtenberg, A. J. and Lieberman, M. A. "Bifurcation Phenomena and Transition to Chaos in Dissipative Systems." Ch. 7 in Regular and Chaotic Dynamics, 2nd ed. New York: Springer-Verlag, pp. 457-569, 1992.

Rasband, S. N. "Asymptotic Sets and Bifurcations." §2.4 in Chaotic Dynamics of Nonlinear Systems. New York: Wiley, pp. 25-31, 1990.

Weisstein, E. W. "Books about Chaos." http://www.ericweisstein.com/encyclopedias/books/Chaos.html.

Wiggins, S. "Local Bifurcations." Ch. 3 in Introduction to Applied Nonlinear Dynamical Systems and Chaos. New York: Springer-Verlag, pp. 253-419, 1990.