Periodic points of complex quadratic mappings
definition
Z_{p} = periodic points of Complex quadratic (Mandelbrot) map when :
Z_{p} = F_{c}^{(n)}(Z_{p})
where n = period
so they are roots of polynomials ( = solutions of polynomial equations) of degree 2 * n
F_{c}^{(n)}(Z_{p})  Z_{p} = 0
As Brian Towles explains on sci.fractals :
F(z) = z [equation for points with period 1 which is 2nd order polynomial ]
F(F(z)) = z [equation for points with period 2, 4th order polynomial ]
.
F(F(F(...F(z)...)))) = z [equation for period n, 2^n order polynomial ]
Only polynomial equations of degree 4 ( quartic ) or less can be solved explicitly.

Fixed points ( period = 1 ) :
 There are 2 (finite) fixed points of F_{c}(z) :
Z_{F1}=0.5sqrt(0.25c)
Z_{F2}=0.5+sqrt(0.25c)

Z_{F1} < Z_{F2}

How to compute fixed points for given c ?
They are solutions of equation:
F_{c}^{(1)}(Z)= Z
z^{2}+c=z
so one gets quadratic equation:
z^{2}z+c:=0
and (finite) fixed points are roots of second degree polynomial:
z^{2}z+c
Compare it with general form of quadratic polynomial:
Az^{2}+Bz+C
so coefficients are:
A:= 1
B:= 1
C:= c = c.x + c.y*i
One can solve it using quadratic formula and rules for operations on complex numbers
How to compute it in delphi
In Maxima :
first method:
(%i1) P(n):=if n=0 then z else P(n1)^2+c;
(%o1) P(n):=if n=0 then z else P(n1)^2+c
(%i10) solve([P(1)z=0], [z]);
(%o10) [z=(sqrt(14*c)1)/2,z=(sqrt(14*c)+1)/2]
second method :
definition:
F[n, c, z] :=
if n=0
then z else (F[n1, c, z]^2 + c);
(%i5) solve([F[1, c, 0]=0], [c]);
(%o5) [c=0]
One can check the results using Viete's formulas:
Z_{F1}+Z_{F2}=B/A= 1
Z_{F1}*Z_{F2}=C/A = c

Stability of fixed points:
Because
Z_{F1}+Z_{F2}= 1
and
dFc / dZ = F'c(Z)= 2*Z
then
F'c(Z_{F1}) + F'c(Z_{F2})= 2*Z_{F1} + 2*Z_{F2}= 2*(Z_{F1}+Z_{F2})=2
so
F'c(Z_{F1}) + F'c(Z_{F2})= 2
This shows that mandelbrot function can have at most 1 attractive finite fixed point.
It is Z_{F1}=0.5sqrt(0.25c)

Parametrization of main component in cplane :
( it is a set of point c for which mandelbrot map has an attractive fixed point )
Periodic point is attractive if stability index is < 1
Abs(F'c(Z_{F1}))< 1
2 * Z_{F1} < 1
2 *  Z_{F1} < 1
 Z_{F1} < 1 / 2
Z_{F1} = (e^{i*angle})/2
However, from
Z_{F1}^{2}Z_{F1}+c:=0
c:= Z_{F1}  Z_{F1}^{2}
so we are looking for c such :
abs(c) < abs ((e ^{i*angle})/2  ( e^{2*i*angle})/4 )
c (angle, radius) = (radius * e ^{i*angle})/2  (radius ^{2} * e^{2*i*angle})/4
where 0< = radius < =1
Boundary of set of such points c is cardioid and angle is internal angle
Parametric definition of border of main cardioid with angle as a parameter:
c(angle) = (e ^{i*angle})/2  (e^{2*i*angle})/4
or
c.x := cos(angle)/2  cos(2*angle)/4
c.y := sin(angle)/2  sin(2*angle)/4
where c:= c.x + c.y * i

 Infinite fixed point = Point at infinity in the Riemann sphere. Any polynomial has a superattractive fixed point at infinity
 period 2 points
{c: 1+c < 1 /4 }
in maxima:
(%i13) solve([P(2)z], [z]);
z=(sqrt(4*c3)+1)/2
z=(sqrt(4*c3)1)/2
z=(sqrt(14*c)1)/2
z=(sqrt(14*c)+1)/2
 period 3 points
 period >3 points
see also:
Main page
Autor: Adam Majewski
adammaj1ato2dotpl
Feel free to email me. (:))
About
republika.pl/fraktal