# Period doubling bifurcations

## Period doubling bifurcation for real quadratic maps

For *c < 1/4*, after the tangent bifurcation,
the *x*_{1} fixed point of the quadratic map (the left
intersection of *f*_{c} and the green line) stays attracting
while its multiplier
*|λ*_{1} | < 1 .
For *c < -3/4* it is *λ*_{1} =
1 - (1 - 4c)^{½} < -1 so the fixed point becomes repelling
and an attracting period-2 orbit appears.
This phenomenon is called the *period doubling bifurcation*.
To watch the birth of the period-2 orbit consider the two-fold iterate
*f*^{ o2}(x) = f(f(x)) of the quadratic map.
It is evident that two fixed points of *f*_{c} are the fixed
points of *f*_{c}^{o2}. Moreover as since for
these points *(f*_{c}^{o2})' =
(f_{c}')^{2}, therefore for *-3/4 < c < 1/2*
the left fixed point of *f*_{c}^{o2} is attracting
(to the left below). At *c = -3/4* it loses stability
and two new intersections of *f*_{c}^{o2} with
the green line simultaneously appear (see to the right). These
*x*_{3 }, x_{4} points make a period-2 orbit.
When this cycle appears *f*^{ o2}(x_{3 })' =
f^{ o2}(x_{4 })' = 1 and the slope decrease as *c*
is decreased.
A period doubling bifurcation is also known as a *flip bifurcation*,
as since the period two orbit flips from side to side about its period
one parent orbit. This is because *f ' = -1* .
At *c =-5/4* the cycle becomes unstable and a stable period-4 orbit
appears. Period doubling bifurcation is called also the *pitchfork
bifurcation* (see below).

## Period doubling bifurcation on complex plane

These pictures illustrate this process on complex plane.
While *c* is changed from *c = 0* to *c = -3/4*
(inside the main cardioid) attractor *z*_{1}
moves from *0* to the parabolic point *p = -1/2* (with
multiplier *λ = -1* ). Two points of
an unstable period 2 orbit are

*z*_{3,4} = -1/2 ± i t,
t = (3/4 + c)^{½} > 0.

Therefore they move towards the point *p* too from above and below.
At *c = -3/4* attractor meets
the repelling orbit and they merge into one parabolic point.
For *c < -3/4*, since *c* leaves the main cardioid
and gets into the biggest *(1/2)* bulb, attractor turns into repeller
and the unstable period-2 orbit becomes attracting with two points

*z*_{3,4} = -1/2 ± t,
t = (-3/4 - c)^{½}.
The right picture above is disconnected Cantor dust. We get it if we
will go up after crossing the *p* point.

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*updated* 12 Sep 2013