Tangent bifurcations

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. Click the "Demo" button to see how 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 (MW).
Bifurcation diagram below shows orbits of the critical point zo = 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 Fig.1.

Birth of period-3 orbits

There are eight points of intersection in the left picture below. Two of them are the period one fixed points, while the remaining six make up a pair of period three orbits. Near to tangency, the absolute value of the slope at three of these points is greater than one - this is the unstable period three orbit. The remaining three points form the stable periodic orbit. In the period-3 window stable and unstable period-3 orbits appear at the tungent bifurcation point c = -1.75 of the f o3 map (at the top of the right picture).
The tungent bifurcation is important since it is one of the most basic processes by which periodic orbits are created.

Tangent bifurcation on complex plane

For quadratic mapping f we have two fixed points z* = f(z*)  
    z1,2 = 1/2 -+ (1/4 - c)1/2,     l1,2 = 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 (i.e. z2 is repeller).

While parameter c belongs to the main cardioid on the scheme below, the fixed point z1 lies into the blue circle and is attracting. Corresponding 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, l1 = 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,   z1, 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 l = 1. For c > 1/4, as c leaves the main cardioid, we get two complex repelling fixed points
    z1,2 = 1/2 -+ t i,   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. For c = 0 (the first image) attractor "A" lies in the center of circles. Repeller "R" always lies in J ! On the second picture you see infinite sequence of preimages of the attractor, repeller and critical point (the critical point "C" lies in the center of J)
Attractor and repeller meet together (above) and we get two repellers. There is disconnected Cantor dust below. Blue points go to Infinity now! Repellers with multipliers m = 1 -+ 2t i generate logarithmic spirals in opposite directions around themselves.

Animations 350+350 and 350x450 pixels. You will see here the play of two repellers (and their preimages) near the point c = 1/4.

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updated 15 August 2003