# Attracting Fixed point and Period 2 orbit

## The main cardioid equation

There are always two fixed points
*z*_{∗} = f(z_{∗ }) for a quadratic map

*f(z*_{∗ }) - z_{∗} =
z_{∗}^{2} + c - z_{∗} = 0,
(1)

*z*_{1,2} = 1/2 ∓ (1/4 - c)^{½}.
(2)

A fixed point with multiplier *λ = f '(z*_{∗ }) =
2z_{∗} is attracting if

*|λ| < 1,
|z*_{∗}| < 1/2,

i.e. *z*_{∗} lies inside the
*u = ½ exp(iφ)* circle. The multiplier on the circle is

*λ = exp(iφ)*. (3)

It follows from (1) that *c = z*_{∗} -
z_{∗}^{2}
and corresponding *c* lies inside the cardioid

*c = u - u*^{2} = ½ exp(iφ)
- ¼ exp(2iφ),

Re(c) = ½ cos(φ) - ¼ cos(2φ),

Im(c) = ½ sin(φ) - ¼ sin(2φ).
## The M-set in the "quadratic" parametrization

We get one more useful "quadratic" parametrization if we use

*c = 1/4 - a*^{2}. (4)

As since *a*^{2} = (-a)^{2}
the **M** is symmetric with respect to *a = 0*. After substitution of
(4) into (2) we get:

*z*_{∗} = 1/2 ± a.

*z*_{∗} is attracting if *|1/2 ± a| < 1/2*,
i.e. *a* lies inside one of the circles

*½ e*^{iφ} ± 1/2 .
So the (4) transformation converts the
main cardioid in two circles.

## Internal angles theory

From (3) it follows that if a fixed point lies at
the *u = ½ exp(i2πm/n)* value then
under iterations its neighbourhood is rotated by
the *φ = 2πm/n* "internal angle".
On the main cardioid the corresponding point lies near
the *m/n* bulb at

*c*_{φ} =
½ e^{iφ} -
¼ e^{2iφ} .

In the "quadratic" parametrization

*a*_{φ} =
½ e^{iφ} - 1/2 .

Therefore *a*_{φ} lays on the
*r = 1/2* circle at the angle *φ*
with respect to the real axis.
## Period 2 orbit

The equation for the period 2 orbit *z*_{o} =
f^{ o2}(z_{o }) = f(f(z_{o })) is

*(z*_{o}^{2} + c)^{2} + c -
z_{o} = (z_{o}^{2} + c -
z_{o })(z_{o}^{2} + z_{o} + c + 1) = 0.

The roots of the first factor are the two fixed points
*z*_{1,2 }. They are repelling outside the main cardioid.
The second factor has two roots

*z*_{3,4} = -1/2 ± (-3/4 - c)^{½}.

These two roots form period-2 orbit.
Since *z*_{3} z_{4} = c + 1 the multiplier of the orbit is

*λ = f '(z*_{3 }) f '(z_{4 }) =
4z_{3} z_{4} = 4(c + 1).

Therefore the orbit is attracting while *|c + 1| < 1/4*
or *c* lies within the *¼ exp(iφ) - 1*
circle. This is exactly equation of the biggest *1/2* bulb to the left
of the main cardioid.
I.e. the main cardioid and the *1/2* bulb are connected and touch each
other in one point *z = -3/4*.

You see the points *z*_{1-4} positions
for *c = -0.71+0.1i* (inside the main cardioid).
Two roots *z*_{3} , z_{4} are
symmetrical with respect to the point *z = -1/2*.
Repeller *z*_{2} lies in Julia set.
Is it "very often" the extreme right point for connected **J**s
("very often" because it is not true e.g. for "cauliflower").

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*updated* 12 Sep 2013