Fold Bifurcation
Let f:R×R->R be a one-parameter family of C^2 map satisfying
![画像: f(0,0)=0 [(partialf)/(partialx)]_(mu=0,x=0)=0 [(partial^2f)/(partialx^2)]_(mu=0,x=0)>0 [(partialf)/(partialmu)]_(mu=0,x=0)>0,](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/FoldBifurcation/NumberedEquation1.svg){kind=link}
then there exist intervals (mu_1,0), (0,mu_2) and epsilon>0 such that
1. If mu in (mu_1,0), then f_mu(x) has two fixed points in (-epsilon,epsilon) with the positive one being unstable and the negative one stable, and
2. If mu in (0,mu_2), then f_mu(x) has no fixed points in (-epsilon,epsilon).
This type of bifurcation is known as a fold bifurcation, sometimes also called a saddle-node bifurcation or tangent bifurcation. An example of an equation displaying a fold bifurcation is
| x^.=mu-x^2 |
(Guckenheimer and Holmes 1997, p. 145).
See also
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References
Guckenheimer, J. and Holmes, P. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, 3rd ed. New York: Springer-Verlag, pp. 145-149, 1997.Rasband, S. N. Chaotic Dynamics of Nonlinear Systems. New York: Wiley, pp. 27-28, 1990.Referenced on Wolfram|Alpha
Fold BifurcationCite this as:
Weisstein, Eric W. "Fold Bifurcation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FoldBifurcation.html