Fractals/Iterations in the complex plane/Mandelbrot set interior

This book shows how to code different algorithms for drawing parameter plane (Mandelbrot set ) for complex quadratic polynomial.

One can find different types of points / sets on parameter plane.

This page is about interior points of the Mandelbrot set.

=Interior of Mandelbrot set - hyperbolic components=

The “Capture-Time” Algorithm: Iterations needed to Converge
The “capture-time algorithm” is a natural counterpart for points inside the set to the “escape-time algorithm”. Given some desired tolerance, the orbit P is generated for each point c ∈ C until some point in the orbit is closer than to some previous point in the orbit. The number of iterations needed for this to occur is mapped to a color and displayed at the pixel corresponding to c. Adam Cunningham

The Lyapunov exponent
Math equation :

$$\lambda_f(z_0) = \lim_{n\rightarrow\infty} \frac{1}{n} \sum_{i=0}^{n-1} \left ( \ln\left|f'(z_i)\right | \right )$$

where:

$$f'(x) = \frac{d}{dz}f_c(z) = 2z$$

means first derivative of f with respect to z

See also:
 * image and description by janthor
 * image by Anders Sandberg

HLSL code by JPBotelho

Interior distance estimation


DEM/M - description of the method

internal level sets
Color of point:
 * is proportional to the value of z is at final iteration.
 * shows internal level sets of periodic attractors.

bof60
Image of bof60 in on page 60 in the book "the Beauty Of Fractals".Description of the method described on page 63 of bof. It is used only for interior points of the Mandelbrot set.

Color of point is proportional to:
 * the smallest distance of its orbit from origin
 * the smallest value z gets during iteration
 * illuminating the closest approach the iterates of the origin (critical point) make to the origin inside the set
 * "Each pixel of each particular video frame represents a particular complex number c = a + ib. For each sequential frame n, the magnitude of z(c,n) := z(c, n-1)^2 + c is displayed as a grayscale intensity value at each of these points c: larger magnitude points are whiter, smaller magnitudes are darker. As n rises from 1 to 256, points outside the Mandelbrot Set quickly saturate to pure white, while points within the Mandelbrot Set oscillate through the darker intensities." Brian Gawalt

Level sets of distance are sets of points with the same distance


 * fragment of code : fractint.cfrm from Gnofract4d

bof60 { init: float mag_of_closest_point = 1e100 loop: float zmag = |z| if zmag < mag_of_closest_point mag_of_closest_point = zmag endif final: #index = sqrt(mag_of_closest_point) * 75.0/256.0 }

See also
 * Mandelbrot Trajectory Infima by jeremy-rifkin Jeremy Rifkin
 * Displaying the Internal Structure of the Mandelbrot Set by Adam Cunningham

bof61 or atom domains
Full description

Period of hyperbolic components
Period of hyperbolic component of Mandelbrot set is a period of limit set of critical orbit.

Algorithms for computing period:
 * direct period detection from iterations of critical point z = 0.0 on dynamical plane
 * "quick and dirty" algorithm : check if $$abs(z_n ) < eps  $$ then colour c-point with colour n. Here n is a period of attracting orbit and eps is a radius of circle around attracting point = precision of numerical computations
 * "methods based on interval arithmetic when implemented properly are capable of finding all period-n cycles for considerable large n." (ZBIGNIEW GALIAS )
 * Floyd's cycle-finding algorithm
 * the spider algorithm
 * atom domain, BOF61
 * Period detection

interior detection
Pixel is interior with high probability if all below is
 * pixel is marked as interior ( black)
 * all surrounding pixels are marked as interior ( black)
 * all the black pixels have the same period

internal coordinate and multiplier map
definition
 * interior coordinate

The algorithm by Claude Heiland-Allen:
 * check c
 * When c is outside the Mandelbrot set
 * give up now
 * or use external coordinate
 * when c is not outside (inside or on the boundary) : For each period p, starting from 1 and increasing:
 * Find periodic point z0 such that fp(z0,c)=z0 using Newton's method in one complex variable
 * Find b by evaluating first derivative with respect to z of fp at z0
 * If |b|≤1 then return b, otherwise continue with the next p

computing
For periods:
 * 1 to 3 explicit equations can be used
 * >3 it must be find using numerical methods

period 1
Start with boundary equation:

c+(w/2)^2-w/2=0;

and solve it for w

(%i1) eq1:c+(w/2)^2-w/2=0; 2                                                                                                            w    w (%o1)                                                                                                        -- - - + c = 0 4   2 (%i2) solve(eq1,w); (%o2)                                                                                       [w = 1 - sqrt(1 - 4 c), w = sqrt(1 - 4 c) + 1] (%i3) s:solve(eq1,w); (%o3)                                                                                       [w = 1 - sqrt(1 - 4 c), w = sqrt(1 - 4 c) + 1] (%i4) s:map(rhs,s); (%o4)                                                                                           [1 - sqrt(1 - 4 c), sqrt(1 - 4 c) + 1]

so

w = w(c) = 1.0 - csqrt(1.0-4.0*c)

period 2
w = 4.0*c + 4;

period 3
$$ c^3 + 2c^2 - (w/8-1)c + (w/8-1)^2 = 0$$

It can be solved using Maxima CAS:

(%i1) e1:c^3 + 2*c^2 - (w/8-1)*c + (w/8-1)^2 = 0;

3     2        w       w     2 (%o1)               c  + 2 c  + (1 - -) c + (- - 1)  = 0 8      8 (%i2) solve(e1,w); (%o2) [w = (- 4 sqrt((- 4 c) - 7) c) + 4 c + 8, w = 4 sqrt((- 4 c) - 7) c + 4 c + 8]

numerical approximation
See also:
 * Number Sequences in the Mandelbrot Set by the Mathemagicians' Guild

Internal angle
Method by Renato Fonseca : "a point c in the set is given a hue equal to argument

$$arg(z_{n_{max}}) = arctan\frac{Im(z_{n_{max}})}{Re(z_{n_{max}})}$$

(scaled appropriately so that we end up with a number in the range 0 - 255). The number z_nmax is the last one calculated in the z's sequence."

See also:
 * [Extra Visual Building a Mandelbrot Set Step-by-step by The Mathemagicians' Guild]

Fractint
Fractint : Color Parameters : INSIDE=ATAN

colors by determining the angle in degrees the last iterated value has with respect to the real axis, and using the absolute value. This feature should be used with periodicity=0

Internal rays
From Hyperbolic to Parabolic Parameters along Internal Rays

When $$radius\,$$ varies and $$angle\,$$ is constant then $$c\,$$ goes along internal ray. It is used as a path inside Mandelbrot set.

Example: internal ray of angle = 1/6 of main cardioid.

Internal angle:

$$angle = 1/6 \,$$

radius of ray:

$$ 0 \le radius \le 1 \,$$

Point of internal radius of unit circle:

$$ w = radius * e^{i * angle}\,$$

Map point $$w$$ to parameter plane:

$$c = \frac{w}{2} - \frac{w^2}{4} \,$$

For $$epsilon = 0 \,$$ this is equation for main cardioid.

Internal curve
When $$radius\,$$ is constant varies and $$angle\,$$ varies then $$c\,$$ goes along internal curve.

Centers of components

 * definition
 * Methods of finding centers

=More tutorials and code=

=References=