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Flip Bifurcation

Let $f:\Bbb{R}\times\Bbb{R}\to\Bbb{R}$ be a one-parameter family of $C^3$ maps satisfying

$\displaystyle f(0,0)$ $\textstyle =$ $\displaystyle 0$  
$\displaystyle \left[{\partial f\over\partial x}\right]_{\mu=0, x=0}$ $\textstyle =$ $\displaystyle -1$  
$\displaystyle \left[{\partial^2 f\over\partial x^2}\right]_{\mu=0, x=0}$ $\textstyle <$ $\displaystyle 0$  
$\displaystyle \left[{\partial^3 f\over\partial x^3}\right]_{\mu=0, x=0}$ $\textstyle <$ $\displaystyle 0.$  

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


See also Bifurcation


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

© 1996-9 Eric W. Weisstein