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)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 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).

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 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 ± 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
    z3,4 = -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