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.
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
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 (i.e. z2 is
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,
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. 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 ∓ 2it generate logarithmic
spirals in opposite directions around themselves.
Animations 350+350 and
You will see here the play of two repellers (and their preimages) near
the point c = 1/4.
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updated 12 Sep 2013