Period doubling bifurcations
Period doubling bifurcation for real quadratic
For c < 1/4, after the tangent bifurcation,
the x1 fixed point of the quadratic map (the left
intersection of fc and the green line) stays attracting
while its multiplier
|λ1 | < 1 .
For c < -3/4 it is λ1 =
1 - (1 - 4c)1/2 < -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 fc are the fixed
points of fco2. Moreover as since for
these points (fco2)' =
(fc')2, therefore for -3/4 < c < 1/2
the left fixed point of fco2 is attracting
(to the left below). At c = -3/4 it loses stability
and two new intersections of fco2 with
the green line simultaneously appear (see to the right). These
x3, x4 points make a period-2 orbit.
When this cycle appears f o2(x3)' =
f o2(x4)' = 1 and the slope decrease as c
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
Pictures below illustrate this process on complex plane
"the birth" scheme
While c is changed from c = 0 to c = -3/4
(inside the main cardioid) attractor z1
moves from 0 to the parabolic point p = -1/2 (with
multiplier λ = -1 ). Two points of
an unstable period 2 orbit are
z3,4 = -1/2 +- t i,
t = (3/4 + c)1/2 (t is real and
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.
Further, for c < -3/4, since c leaves the main cardioid,
attractor turns into repeller and as c gets into the biggest (1/2)
bulb the unstable period-2 orbit becomes attracting with two points
z3,4 = -1/2 +- t,
t = (-3/4 - c)1/2.
The right picture above is disconnected Cantor dust. We get it if we
will go up after crossing the p point.
350+450 pixels movies)
Previous: Tangent bifurcations
Next: Period trippling bifurcations
updated 12 June 2003