Period doubling bifurcations

Period doubling bifurcation for real quadratic maps

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 1 | < 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 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).
Tangent

Period doubling bifurcation on complex plane

Pictures below illustrate this process on complex plane Scheme "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 positive).
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.

Animation   (350+350 and 350+450 pixels movies)


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updated 12 June 2003