4.6: Period doubling bifurcations

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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 f_c and the green line) stays attracting while its multiplier |λ₁ | < 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 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 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

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.