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Bifurcation


BifurcationLogistic

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 r is varied. Bifurcations come in four basic varieties: flip bifurcation, fold bifurcation, pitchfork bifurcation, and transcritical bifurcation (Rasband 1990).

BifurcationBranches

More generally, a bifurcation is a separation of a structure into two branches or parts. For example, in the plot above, the function R[sqrt(z^2)], where R[z] denotes the real part, exhibits a bifurcation along the negative real axis x=R[z]<0 and y=I[z]=0.


See also

Branch, Codimension, Feigenbaum Constant, Feigenbaum Function, Flip Bifurcation, Fold Bifurcation, Hopf Bifurcation, Logistic Map, Period Doubling, Pitchfork Bifurcation, Transcritical Bifurcation

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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.

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Bifurcation

Cite this as:

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

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