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
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.
Tangent bifurcation on complex plane
For the quadratic mapping f we have two fixed points
z1,2 = 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,
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 λ =
1. For c > 1/4, as c leaves the main cardioid,
we get two complex repelling fixed points
z1,2 = 1/2 ∓ i t,
t = (c - 1/4)1/2
where t is real. They go away the p point in the vertical
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.
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.
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updated 12 Sep 2013