# 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. 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 *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.
## 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},
λ_{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.

While parameter *c* belongs to the main cardioid on the scheme,
the fixed point *z*_{1} lies into the blue circle and is
attracting and 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,
λ_{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 *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 *z*_{1}.
On the second picture you see infinite sequence
of preimages of the attractor, repeller and critical point.

*"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.

Contents
Previous: Attracting fixed point and period 2 orbit
Next: Birth of Attracting period 2 orbit

*updated* 12 Sep 2013