Periodic_points

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=0.5-sqrt(0.25-c)
  Z_F2=0.5+sqrt(0.25-c)
  ---------
  Z_F1 < Z_F2
  
  ----------------------------
  How to compute fixed points for given c ?
  They are solutions of equation:
  F_c⁽¹⁾(Z)= Z
  z²+c=z
  so one gets quadratic equation:
  z²-z+c:=0
  and (finite) fixed points are roots of second degree polynomial:
  z²-z+c
  Compare it with general form of quadratic polynomial:
  Az²+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(n-1)^2+c;
  (%o1) P(n):=if n=0 then z else P(n-1)^2+c
  (%i10) solve([P(1)-z=0], [z]);
  (%o10) [z=-(sqrt(1-4*c)-1)/2,z=(sqrt(1-4*c)+1)/2]
  second method :
  definition:
  F[n, c, z] :=
  if n=0
  then z else (F[n-1, 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=0.5-sqrt(0.25-c)
  -----------------------------------------------------
  Parametrization of main component in c-plane :
  ( 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_F1²-Z_F1+c:=0
  c:= Z_F1 - Z_F1²
  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*c-3)+1)/2
z=(sqrt(-4*c-3)-1)/2
z=-(sqrt(1-4*c)-1)/2
z=(sqrt(1-4*c)+1)/2
period 3 points
period >3 points