Attracting Fixed point and Period 2 orbit
The main cardioid equation
We have seen before that there are always two fixed points
z∗ = f(z∗ ) for a quadratic map
f(z∗ ) - z∗ =
z∗2 + c - z∗ = 0,
z1,2 = 1/2 ∓ (1/4 - c)½.
Since multiplier λ = f '(z∗ ) =
2z∗ a fixed point is attracting if
|λ| < 1,
|z∗| < 1/2
i.e. z∗ lies inside the
u = ½ exp(iφ) circle.
It follows from (*) that c = z∗ - z∗2
and corresponding c lies inside the cardioid
c = u - u2 = 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 "square" parametrization
We get one more usefull "square" parametrization if we use
c = 1/4 - a2. (**)
As a2 = (-a)2
the M is symmetrical with respect to a = 0. After substitution of
(**) into (*) 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 eiφ ± 1/2 .
So (**) transformation converts the
main cardioid in two circles.
Internal angles theory
Point on the main cardioid corresponding to an internal angle
φ = 2πm/n lies at
½ eiφ -
¼ e2iφ .
In "square" parametrization
½ eiφ - 1/2 .
Therfore aφ lays on the
r = 1/2 circle at the angle φ
with respect to the real axis.
Period 2 orbit
Equation for period 2 orbit zo = f o2(zo ) =
f(f(zo )) is
(zo2 + c)2 + c -
zo = (zo2 + c -
zo )(zo2 + zo + c + 1) = 0.
The roots of the first factor are the two fixed points
z1,2 . They are repelling outside the main cardioid.
The second factor has two roots
z3,4 = -1/2 ± (-3/4 - c)½.
These two roots form period-2 orbit.
Since z3 z4 = c + 1 the multiplier of the orbit is
λ = f '(z3 ) f '(z4 ) =
4z3 z4 = 4(c + 1).
Therefore the orbit is attracting while |c + 1| < 1/4
or c lies within the [exp(iφ) / 4 - 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 z1-4 positions
for c = -0.71+0.1i (inside the main cardioid).
Two roots z3 , z4 are
symmetrical with respect to the point z = -1/2.
We will watch fixed points and periodic orbits movement in more detail on
the next pages.
Repeller z2 lies in Julia set.
Is it "very often" the extreme right point for connected Js
("very often" because it is not true e.g. for "cauliflower").
Previous: Spiral structures in the Julia sets
Next: Tangent bifurcations
updated 12 Sep 2013