4.5: Tangent bifurcations

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The left intersection of the green line and parabola is an attracting fixed point because the absolute value of the f(x) slope at the point is smaller than one. The slope at the right intersection is greater than one and it is a repeller. These points meet together at c = 1/4. For c > 1/4 the fixed points become complex and repelling. This is the tangent (or fold) bifurcation.

Bifurcation diagram below shows orbits of the critical point z_o = 0. You see filaments (and broadening) which show, how iterations converge to the attracting fixed point z₁. It is superattracting for c = 0. For c > 1/4 (at the top of the picture) iterations go away to infinity. Repelling fixed point z₂ created at the tangent bifurcation is shown in \(\PageIndex{2}\) .

Tangent bifurcation on complex plane

For the quadratic mapping f we have two fixed points

z_1,2 = 1/2 ∓ (1/4 - c)^1/2, λ₁_,2 = 2z₁_,2 .

Since z₂ = 1/2 - z₁ the roots are situated symmetrically with respect to the point p = 1/2. The square root function maps the whole complex plane into a complex half-plane. We choose the Re(z) > 0 half-plane here, therefore the fixed point z₂ is always repelling.

While parameter c belongs to the main cardioid on the scheme, the fixed point z₁ lies into the blue circle and is attracting and repeller z₂ lies into the yellow circle. z₁ becomes a repeller too when c lies outside the main cardioid.

For c = 0 we have z₁ = 0, λ₁ = 0 that is z₁ lies in the center of the blue circle and is a superattracting point. z₂ lies in the center of the yellow circle. While c goes to 1/4, z_{1, 2} move towards the p point. For c = 1/4 the two roots merge together in p and we get one parabolic fixed point with multiplier λ = 1. For c > 1/4, as c leaves the main cardioid, we get two complex repelling fixed points

z_1,2 = 1/2 ∓ i t, t = (c - 1/4)^1/2

where t is real. They go away the p point in the vertical direction.

On the pictures below colors inside filled J sets show how fast a point goes to attractor z₁. On the second picture you see infinite sequence of preimages of the attractor, repeller and critical point.

4.5.5.jpg — Figure \(\PageIndex{5}\): *"cauliflower"*

When attractor and repeller meet together we get two repellers. There is disconnected Cantor dust below. Blue points go to Infinity now! Repellers with multipliers λ = 1 ∓ 2it generate logarithmic spirals in opposite directions around themselves.