Fractals/Mathematics/Derivative

• Derivative of Iterated function (map)
• of the function f with respect to variable ...
• following the derivative
• the rolling derivative
• the derivative of order n: first, second, ...

notation

• variable z
• constant parameter c
• function f
• derivative d
• iteration number p
• wrt = with respect to

Notation types

• Leibniz's notation
• Euler's notation
• Newton's notation ( dot notation)
• Lagrange's notation ( prime mark notation)

${\frac {d}{dz}}f(z)=f'(z)=z'=d=D\,$

Derivative of the iterated function

  "the derivative basically as it's calculated for anlytical DE using the implementation of the chain rule for the derivative of f(g(x)) where f(x) is say g(x)^p+c and the value of g(x) is current z. So f'(g(x)) is p*z^(p-1)  and g'(x) is the derivative from the previous iteration,  so on each iteration the new derivative is:  dz = dz*p*z^(p-1)  and new z = z^p+c as normal" FractalDave

// initial values:
z = z
d = 1


// Recurrence steps: 
d = f'(z)*d   // calculating d and multiply the previous d
z = f(z)       // forward iteration

It can be computed with Maxima CAS :

display2d:false;
f:1/(z^3+d*z);
dz : diff(f,z,1); // first derivative of f wrt variable z 
dc :  diff(f,c,1); // first derivative of f wrt parameter c

Rational function

 $f(x)={\frac {P(x)}{Q(x)}}$

 $f'(x)={\frac {P'(x)Q(x)-P(x)Q'(x)}{Q^{2}(x)}}$

degree 3

wrt z

// function 
f(z)= 1/(z^3 + c*z )

// first derivativwe wrt z
d = f'(z) = -(3z^2 +c)/(z^3 + cz)^2

// iteration:
z_(n+1) = f(z_n)


// initial values:
z = z
d = 1

// Recurrence steps:
d = - d*(3z^2 +c)/(z^3 + cz)^2
z = 1/(z^3 + c*z) 

degree 5

Julia set f(z)=1 over az5+z3+bz

display2d:false;
f: 1/(a*z^5+ z^3 + b*z);
dz: diff(f,z,1);

in plain text:

  dz = -(5*a*z^4+3*z^2+b)/(a*z^5+z^3+b*z)^2

Example:

• 1/((0.15*I+0.15)*z^5+(3*I-3)*z + z^3)

degree 5 by Buschmann

Julia set drawn by distance estimation, the iteration is of the form $f_{c}(z)=1-z^{2}+{\frac {z^{5}}{2+4z}}+c$ in black and white

f(z):=z^5/(4*z+2)-z^2+c+1

(%i4) diff(f(z), z,1);

(%o4) (5*z^4)/(4*z+2)-(4*z^5)/(4*z+2)^2-2*z

Polynomials

Complex quadratic polynomial

First derivative wrt c

On parameter plane :

• $c$ is a variable
• $z_{0}=0$ is constant

${\frac {d}{dc}}f_{c}^{(p)}(z_{0})=z'_{p}\,$

This derivative can be found by iteration starting with

$z_{0}=0\,$

$z'_{0}=1\,$

and then ( compute derivative before next z because it uses previous values of z):

$z'_{p+1}=2\cdot z_{p}\cdot z'_{p}+1\,$

$z_{p+1}=z_{p}^{2}+c\,$

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

${\frac {dz_{n+1}}{dc}}={\frac {d(z_{n}^{2}+c)}{dc}}=2z_{n}{\frac {dz_{n}}{dc}}+{\frac {dc}{dc}}=2z_{n}{\frac {dz_{n}}{dc}}+1\,$

• Maxima CAS function :

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

Example values :

$z_{0}=0\qquad \qquad z'_{0}=1\,$

$z_{1}=c\qquad \qquad z'_{1}=1\,$

$z_{2}=c^{2}+c\qquad z'_{2}=2c+1\,$

It can be used for:

• the distance estimation method for drawing a Mandelbrot set ( DEM/M )

• Mandelbrot set boundary = DEM/M
• 

First derivative wrt z

$z'_{n}\,$ is first derivative with respect to z.

This derivative can be found by iteration starting with

$z_{0}=z\,$

$z'_{0}=1\,$

and then :

$z'_{n+1}=2*z_{n}*z'_{n}\,$

$z_{n+1}=z_{n}^{2}+c\,$

description	arbitrary name	formula	Initial conditions	Recurrence step	common names
iterated complex quadratic polynomial	$A_{m}$	$f^{m}$	$A_{0}=z$	$A_{m+1}=A_{m}^{2}+c$	z or f
first derivative with respect to z	$B_{m}$	${\dfrac {\partial }{\partial z}}f^{m}$	$B_{0}=1$	$B_{m+1}=2A_{m}B_{m}$	dz, d (from derivative) or p ( from prime) of f'

Derivation of recurrence relation:

$A_{0}\quad {\xrightarrow {definition\ of\ A}}\ f^{0}(z,c)\quad {\xrightarrow {definition\ of\ f}}\ \quad z$

$A_{m+1}\quad {\xrightarrow {definitionofA}}f^{m+1}(z,c)\quad {\xrightarrow {definition\ of\ f}}f(f^{m}(z,c),c)\quad {\xrightarrow {definition\ of\ A}}f(A_{m},c)\quad {\xrightarrow {definition\ of\ f}}A_{m}^{2}+c$

$B_{0}\quad {\xrightarrow {definition\ of\ B}}\ {\dfrac {\partial }{\partial z}}f^{0}(z,c){\xrightarrow {definition\ of\ f}}\ {\dfrac {\partial }{\partial z}}z{\xrightarrow {derivative}}\ 1$

$B_{m+1}\quad {\xrightarrow {definition\ of\ B}}{\dfrac {\partial }{\partial z}}A_{m+1}\quad {\xrightarrow {definition\ of\ A}}{\dfrac {\partial }{\partial z}}(A_{m}^{2}+c)\quad {\xrightarrow {distributivity}}{\dfrac {\partial }{\partial z}}A_{m}^{2}+{\dfrac {\partial }{\partial z}}c\quad {\xrightarrow {constant\ derivative}}{\dfrac {\partial }{\partial z}}A_{m}^{2}+0\quad {\xrightarrow {zero}}{\dfrac {\partial }{\partial z}}A_{m}^{2}\quad {\xrightarrow {chain\ rule}}2A_{m}({\dfrac {\partial }{\partial z}}A_{m})\quad {\xrightarrow {definition\ of\ B}}2A_{m}B_{m}$

Following the derivative

"As we iterate z, we can look at the derivatives of P at z. In our case it is quite simple: P′(z)=2z. Multiplying all these numbers along an orbit yields the derivative at z of the composition Pn. This multiplication can be carried on iteratively as we iterate z " A Cheritat

 der = 1
 z = c # note that we start with c instead of 0, to avoid multiplying the derivative by 0
 for n in range(0,N):
   new_z = z*z+c
   new_der = der*2*z
   z = new_z
   der = new_der

It can be used for :

• computing the external distance to the Julia set (DEM/J)
• detection of interior points of Mandelbrot set componnets

unsigned char ComputeColorOfDEMJ(complex double z){
// https://en.wikibooks.org/wiki/Fractals/Iterations_in_the_complex_plane/Julia_set#DEM.2FJ


  int nMax = iterMax;
  complex double dz = 1.0; //  is first derivative with respect to z.
  double distance;
  double cabsz;
	
  int n;

  for (n=0; n < nMax; n++){ //forward iteration

    if (cabs(z)> 1e60 || cabs(dz)> 1e60) break; // big values
    
  			
    dz = 2.0*z * dz; 
    z = z*z +c ; /* forward iteration : complex quadratic polynomial */ 
  }
  
  cabsz = cabs(z);
  distance = 2.0 * cabsz* log(cabsz)/ cabs(dz);
  if (distance <distanceMax) {//printf(" distance = %f \n", distance); 
  		return iColorOfBoundary;}
  
  
  return iColorOfExterior;
}

• Julia set boundary = DEM/J
• 
• 

logistic map

Logistic map

complex cubic polynomial

wrt z

Julia set for f(z) = z^3 +z*(0.1008317508132964*i + 1.004954206930806)

// function 
f(z)= z^3 + c*z 

// first derivativwe wrt z
d = f'(z) = 3*z^2 + c

// iteration:
z_(n+1) = f(z_n)


// initial values:
z = z
d = 1

// Recurrence steps:
d = (3*z^2 + c)*d
z = z^3 + c*z 

FAQ

• why new d must be computed before new z ?

Aplications

• Vector field gradient
• Newton method and recurrence relations
• DEM = Distance Estimation Method
  • DEM/M- for Mandelbrot set
  • DEM/J for Julia set
• stability of periodic point ( multiplier)
• finding critical points

See also

• automatic differentiation