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I) For the sake of repetition and comparison, let’s start with the
quadratics.
The term “quadratics” refers to 2-degree polynomials, i e when
the highest power of the variable “z” is two. Quadratic polynomials, when viewed
as dynamical systems, are smartest studied in the form of the well-known
formula: z -> z^2 + c, “z” and “c” complex. The dynamical plane is made up of
z_real and z_imag. There are two kinds of regions in the dynamical plane.
1) Those regions that have “z” those orbits escape to infinity under
iteration. The pixels corresponding to those “z” are coloured according to the
number of iterations required to take the orbits of those “z” beyond a certain
radius.
2) Those regions that have “z” those orbits are forever bounded within a
small radius from the centre of the dynamical plane. The pixels corresponding to
those “z” are coloured in one colour, usually black
(if you don’t use
certain filters, in these days so popular).
The Julia set is the border between these regions. On this border there
are “sensitive dependence on initial condition”. The Julia set is therefore also
called the “chaotic set”. The Julia set together with eventually enclosed
regions is called filled in Julia set (and is usually coloured black).
Question: What does a Julia set look like?
Answer: The shape of the Julia set is entirely governed by the
complex parameter “c”. If c = 0 the Julia set is the unit circle. If c = -2 the
Julia set is a straight line between –2 and +2. For all other values of “c”, the
Julia set is a fractal.
Now there are two different kind of quadratic Julia
sets: 1) connected and 2) totally disconnected, made up of so called “Cantor
dust”.
 |
 |
| fig 1 | fig 2 |
Figure 1 is an example of the first and figure 2 of the second.
Answer: Yes, you only have to iterate the critical point. In the first
case the critical point belongs to the filled in Julia set and thus have an
orbit that is bounded forever. In the second case the critical point doesn’t
belong to the filled in Julia set and thus have an orbit that tends to infinity.
Question: How do I get the critical point?
Answer: You put the derivative of your polynomial to zero. In our case we
have p(z) = z^2 + c, the derivative is p’(z) = 2z. 2z = 0 makes the critical
point z = 0. In figures 1 and 2, the critical point is marked with yellow
dots.
| Now we have a beautiful tool to draw the plane of all
parameters “c” which give rise to connected Julia sets. For each c = pixel
we only have to let our computer test if z = 0 has an orbit that escapes
to infinity, in which case we colour the pixel corresponding to that “c”
according to the number of iterations it takes for the critical point (z =
0) to pass a certain radius (which must be at least two). If on the other
hand the critical point (z = 0) has a bounded orbit, we colour the pixel
corresponding to that “c” black. It is those values on the parameter “c”
that constitute the famous Mandelbrot set. As the Mandelbrot set is the set of all parameters “c” that give rise to connected Julia sets, the Mandelbrot set is also called “Quadratic connectedness locus”. Figure 3 shows the quadratic parameter space (=plane). |
 figure 3 |
The left yellow dot, marks the parameter belonging to the Mandelbrot
set,
which gives rise to the connected Julia set in figure 1.
The right yellow dot marks the parameter outside the Mandelbrot
set,
which gives rise to the disconnected Julia set in figure 2.
A very important thing is: The Julia sets resides in the dynamical
plane, i e the z-plane where the iterations take place. The Mandelbrot set on
the other hand, resides in the parameter plane, i e the c-plane.
II) The term “cubics” refers to 3-degree polynomials, i e the highest
power of the variable “z” is three. Cubic polynomials, viewed as dynamical
systems, are smartest studied in the soon well-known form:
z -> z^3 - 3a^2
z + b.
Question: Which funny things happen when we iterate z
-> z^3 - 3a^2 z + b that are not there when we iterate z -> z^2 + c?
Answer: The first funny thing are the critical points. Cubic polynomials
happen to have two of that kind. We simply get them by putting the derivative to
zero, i e:
p’(z) = 3z^2 - 3a^2 = 0
3z^2 = 3a^2
z = +-a
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| fig 4 | fig 5 |
| 1) Both critical points have orbits that tend to
infinity. The resulting Julia set is totally disconnected (figure 4). The
yellow dots in figure 4, 5, and 6 are the critical points. 2) One critical point has an orbit that tends to infinity, and the other critical point has a bounded orbit. This is the new case compared with quadratic polynomials, which only has one critical point. The resulting Julia set is disconnected but only in rare cases the set is a totally disconnected Cantor set. In most cases the Julia set encloses domains where one of the critical points belongs to such a domain. But every such domain is separated from every other domain (figure 5). 3) Both critical points have bounded orbits. The resulting Julia set is connected (figure 6). |
 fig 6 |
The next funny thing is the cubic parameter space. For quadratics, z
-> z^2 + c, there is one parameter “c”. As “c” is complex, we have two axis,
c_real and c_imag, i e the quadratic parameter space is a plane actually. In the
cubic case there are two complex parameters, “a” and “b”, i e there are the axis
a_real, a_imag, b_real, and b_imag.
That means that the cubic parameter space
is a four dimensional hyper space!!! However that’s not a problem. If you’ve
started the sub-module 3D Cubic, you will under “fdim” decide which axis you
want to point into the fourth dimension, and you obtain a 3D slice of this
four-dimensional cubic monster. Then you put a number on “u-coord” in order to
declare where along the axis, pointing in the fourth dimension, you want to make
your 3D- slice. You can also make 2D-slices from this module.
The last picture (figure 7) is a two-dimensional slice
from cubic parameter space, where
a_real = 0.57735, a_imag =
0,
b_real = pixel, and b_imag = pixel.
The darkblue-orange set plus all of the
black is M+, the set for which every z = +a has a bounded orbit. The blue
set plus all of the black is M-, the set for which every z = -a has a
bounded orbit. The black set is the set that is common to both M+ and M-
(the terms M+ and M- are adopted from Science of Fractal Images). This
common set is called “Cubic connectedness locus” or C-locus (ccl in the 3D
module of Stig). In the picture there are three yellow dots. The one
to the right, belonging to neither M+ nor M-, denotes the b-parameter
which together with the above fixed a-parameter, give rise to the totally
disconnected Julia set in figure 4. 
fig 7
The middle one, belonging to M+ but not M-, denotes the b-parameter which together with the above fixed a-parameter, give rise to the disconnected Julia set in figure 5. The one to the left, belonging to both M+ and M- i e C-locus, denotes the b-parameter which together with the above fixed a- parameter, give rise to the connected Julia set in figure 6.
Some more definitions:
That part of cubic parameter space for which both critical points have orbits
those escapes to infinity
is called E-locus (E for escape).
That part of cubic parameter space for which one critical point has an orbits
that escapes to infinity and the other critical point has a bounded orbit, i e
that belongs to either M+ or M- but not both,
is called B-locus (I guess B
stands for boundary).
That part of cubic parameter space for which both critical points have bounded orbits is called C-locus
(C for connected), i e Cubic connectedness locus, the full analogy to the Mandelbrot set for quadratics.
These definitions are adopted from the work of Branner and Hubbard. I adopt the term “Cubics” for this subject from a subheading in “A FIRST COURSE IN CHAOTIC DYNAMICAL SYSTEMS” written by Robert Devaney.
Many fractal people associate” cubics” with the cubic Mandelbrot set,
residing on the c-plane when iterating z -> z^3 + c. In fact the cubic
Mandelbrot set is a special 2D-slice of cubic connectedness locus when a_real
and a_imag are fixed to zero, and the critical points +a and –a thus coalesces
in zero.
The images to this article are produced with Ultra fractal. The cubic
Julia set is produced with the sub-module “Cubic Julia” (In versions
before Aug 1999-08-08 this sub-module contains a mathematical bug.
Please upgrade the sp-module). The figure 7 is produced with the
earlier sub-module “Cubic Parameter-space” which allows you to
investigate all 2D-slices of Cubic Parameterspace. One layer is used
for M+, another for M-. The same 2D-slice can also easily be done with
“3D Cubic”. It’s a matter of taste which module to use for 2D-slices.
The old “Cubic Parameterspace” is a little bit faster, but the new “3D
cubic” is a little bit more pedagogical. Since spring 2000 there is
also a raytracing module for 3D cubics (as well as for Juliabrot,
Quaterions, Hypercomplex etc). The very first picture in this article
is produced with this new module (spr). Many thanks to my dear friend
who has written these wonderful modules, and have put this tutorial to
the web. His latest modules (as well as mine) can be downloaded from here
And here you find Ultra fractal.
A little bit historical site dealing with the method I used in drawing slices from cubic parameter space 1996 – 1997 is: Fractal Holmes Cubics
At last - Terry Gintz have a site, which besides Julias drawn in quaternions, also contains 3D - slices of cubic parameter space.
New! Chaotic series of fractal articles
Regards,
Ingvar Kullberg
Unique visitors since 2010-08-16
Copyright © 1999, 2002 Ingvar Kullberg
Uppdated 2007-02-03