# 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
*z*_{o} = 0. You see filaments (and broadening) which show,
how iterations converge to the attracting fixed point *z*_{1}.
It is superattracting for *c = 0*. For *c > 1/4* (at the top of the
picture) iterations go away to infinity. Repelling fixed point
*z*_{2} 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_{*})

* z*_{1,2} = 1/2 -+ (1/4 - c)^{1/2},
l_{1,2} = 2z_{1,2} .

Since *z*_{2} = 1/2 - z_{1}
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 *z*_{2} is always repelling (i.e. *z*_{2} is
*repeller*).

While parameter *c* belongs to the main cardioid on the scheme below,
the fixed point *z*_{1} lies into the blue circle and is
attracting. Corresponding repeller *z*_{2} lies into the
yellow circle. *z*_{1} becomes a repeller too when *c* lies
outside the main cardioid.

For *c = 0* we have *z*_{1} = 0,
l_{1} = 0
that is *z*_{1} lies in the center of the blue circle and is
a superattracting point. *z*_{2} 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 *l =
1*. For *c > 1/4*, as *c* leaves the main cardioid,
we get two complex repelling fixed points

*z*_{1,2} = 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 *z*_{1}. 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**)

*"cauliflower"*

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*.

Contents
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Next: Birth of Attracting period 2 orbit

*updated* 15 August 2003