Flip Bifurcation

Let $f:\Bbb{R}\times\Bbb{R}\to\Bbb{R}$ be a one-parameter family of 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

$\begin{displaymath} f(x)=\mu-x-x^2. \end{displaymath}$

See also Bifurcation

References

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