Fractals/Iterations in the complex plane/def cqp

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[edit] Definitions

[edit] Complex quadratic map

  • math notation : f_c(z)=z^2+c\,
  • Maxima function :
f(z,c):=z*z+c;

[edit] Iterated function (map)

  • math notation
\ f^{(p)} _c (z) = f_c(f^{(p-1)} _c (z))

where

\ f^{(0)} _c (z) = z
\ f^{(1)} _c (z) = f_c(z)


\ z_p = f^{(p)} _c (z)
  • Maxima notation :
fn(p, z, c) :=
  if p=0 then z
  elseif p=1 then f(z,c)
  else f(fn(p-1, z, c),c);

zp:fn(p, z, c);

[edit] Derivative of Iterated function (map)

[edit] Derivative with respect to c

\frac{d}{dc} f^{(p)} _c (z) = z'_p \,

This derivative can be found by iteration starting with


z_0 = z \,
z'_0 = 1 \,


and then

z_p = z_{p-1}^2 + c \,
z'_p = 2 \cdot z_{p-1}\cdot z'_{p-1} + 1 \,

This can be verified by using the chain rule for the derivative.

Maxima notation :


dcfn(p, z, c) :=
  if p=0 then 1
  else 2*fn(p-1,z,c)*dcfn(p-1, z, c)+1;


Example values :

z_0 = z \qquad z'_0 = 1 \,
z_1 = z^2 + c \qquad z'_1 = 2z + 1 \,
z_2 = z^4+2cz^2+c^2+c \qquad z'_2 = 4z^3+2z^2+4cz+2c+1 \,
z_3 = z^8+4cz^6+6c^2z^4+2cz^4+4c^3z^2+4c^2z^2+c^4+2c^3+c^2+c \qquad z'_3 = 8z^7+4z^6+24cz^5+12cz^4+2z^4+24c^2z^3+8cz^3+12c^2z^2+8cz^2+8c^3z+8c^2z+4c^3+6c^2+2c+1 \,

[edit] Derivative with respect to z

[edit] Standard polynomial F_p \, which roots are periodic z-points of period p and its divisors

  • math notation : \ F_p(z,c) = f^{(p)} _c (z) - z [1]
  • Maxima function :
F(p, z, c) := fn(p, z, c) - z ;

[edit] Function for computing reduced polynomial G_p\, which roots are periodic z-points of period p without its divisors

  • math definition : G_p(z,c) = \frac{F_p(z,c) }{ \prod_{m|p,m<p} G_m(z,c)} \,
  • Maxima function:
G[p,z,c]:=
block(
[f:divisors(p),
t:1], /* t is temporary variable = product of Gn for (divisors of p) other then p */
f:delete(p,f), /* delete p from list of divisors */
if p=1
then return(F(p,z,c)),
for i in f do 
 t:t*G[i,z,c],
g: F(p,z,c)/t,
return(ratsimp(g))
)$

[edit] Multiplier of periodic z-point

\lambda = \frac{df_c^{(p)}(z)}{dz}\,

m(p):=diff(fn(p,z,c),z,1);

[edit] Unit circle \partial D\, = boundary of unit disk

\partial D = \left\{ w: abs(w)=1 \right \}

where coordinates of w\, point of unit circle in exponential form are :

w = e^{i*t}\,

[edit] Hyperbolic component of Mandelbrot set

Domain is an open connected subset of a complex plane.

"A hyperbolic component H of Mandelbrot set is a maximal domain on which f_c\, has an attracting periodic orbit.

A center of a H is a parameter c_0 \in H\, such that the corresponding periodic orbit has multiplier 0." [2]

[edit] References

  1. ↑ Numerical Methods for Finding Periodic Orbits at Scholarpedia
  2. ↑ Surgery in Complex Dynamics by Carsten Lunde Petersen, online paper
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