Flip Bifurcation
Let f:R×R->R be a one-parameter family of C^3 maps satisfying
f(0,0) =
(1)
[画像:[(partialf)/(partialx)]_(mu=0,x=0)] = -1
![画像:[(partialf)/(partialx)]_(mu=0,x=0)](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/FlipBifurcation/Inline6.svg){kind=link}
(2)
![画像:[(partial^2f)/(partialx^2)]_(mu=0,x=0)](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/FlipBifurcation/Inline9.svg){kind=link}
[画像:[(partial^3f)/(partialx^3)]_(mu=0,x=0)] < 0.
![画像:[(partial^3f)/(partialx^3)]_(mu=0,x=0)](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/FlipBifurcation/Inline12.svg){kind=link}
(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
BifurcationExplore with Wolfram|Alpha
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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 BifurcationCite this as:
Weisstein, Eric W. "Flip Bifurcation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FlipBifurcation.html