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Flip Bifurcation


Let f:R×R->R be a one-parameter family of C^3 maps satisfying

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

Then there are intervals (mu_1,0), (0,mu_2), and epsilon>0 such that

1. If mu in (0,mu_2), then f_mu(x) has one unstable fixed point and one stable orbit of period two for x in (-epsilon,epsilon), and

2. If mu in (mu_1,0), then f_mu(x) has a single stable fixed point for x in (-epsilon,epsilon).

This type of bifurcation is known as a flip bifurcation. An example of an equation displaying a flip bifurcation is

 f(x)=mu-x-x^2.
(5)

See also

Bifurcation

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References

Rasband, S. N. Chaotic Dynamics of Nonlinear Systems. New York: Wiley, pp. 27-30, 1990.

Referenced on Wolfram|Alpha

Flip Bifurcation

Cite this as:

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

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