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==== An Aesthetic Exploration of Multivariate Polynomial Maps ====

this research report is still in progress.

==== Context ====

In 2001 I found a book by mathematician Julien C. Sprott entitled "Strange
Attractors: Creating Patterns in Chaos" (CPiC). The book describes the
mathematics behind a class of fractals called strange attractors.

The bulk of CPiC deals with a sub-class of strange attractors called iterated
maps. An iterated map is an equation, or often a system of equations, which are
evaluated iteratively. Iteration in this case means that we take the result of
evaulating the equations with, and feed this back into the same equations as
inputs for the next evaluation.  With most randomly chosen equations, the set
of results produced by succesive iterations is not particularly interesting,
perhaps rapidly climbing to infinity, collapsing to 0 or some other fixed
value. For a small number of equations however, the set of results carves out a
region of number space which exhibits interesting fractal properties. This
region of number space is called an attractor. The complete term "strange
attractor" refers also to the chaotic propeties of these attractors (CPiC p10,
CaTSA p127).

In CPiC Sprott presents a computer program which can search for the equations
which produce strange attractors by trying many randomly generated equations
until one is discovered which meets certain criteria. The results are clouds of
points which formed interesting shapes, many with an organic quality. Each set
of equations created a unique set of points.

The organic quality of the images is particularly intriguing as they are the
result of a relatively simple mathematical process. In fact it is this
complexity, emerging from apparently simple equations, which has attracted
mathematicians.

The book describes a range of different classes of equations which exhibit the
property of producing strange attractors. One class of equations which
particularly caught my attention was the polynomial map.  Polynomials are
interesting as they are easy to generalise, meaning one can easily add or
remove terms from the equation to experiment with systems of greater or lesser
complexity.  Additionally, this flexibility is easily parameterised, making it
possible to automate. Finally, there are a number of properties of polynomials
that makes their evaluation on computers convenient.

With each class of strange attractor producing equations presented in the book,
there were a number of parameters (coefficients, constants) that could be a
adjusted to produce different attractors. Often, only a small percentage of the
possible constant values would give rise to a strange attractor. The program
that Sprott develops in the course of the book randomly tests many different
randomly generated parameters, and displays the strange attractor produced by
those which exhibited favourable properties.  Since the values were chosen
randomly from the infinite range of possible values, it gives the impression of
the different attractors lying spread out discretely across number space,
isolated from each other like stars in the night sky.

I was curious to see what would be the outcome of making small changes to the
chosen values, to see if the attractors did not exist as pinpricks in number
space, but possibly as extended areas, possibly even connected.  I made a
simple program which did this, starting from one known strange attractor and
making small changes to the parameters, and plotting the resulting attractor.
The result was the familiar clouds of points, slowly morphing between many
different shapes.

The simple program I made would move through the parameter space randomly and
automatically. It showed that there was a continuity to the parameter space
which the strange attractors occupied. It did not however, provide any useful
way of visualising the path it was taking through this space.

In this research I would like to explore this space in a more controlled
manner, and possibly map it to some degree. The mapping of fractal spaces is
not unheard of in mathematics. As a simple example, a region of the Lyapunov
fractal has been dubbed "Zircon Zity" by mathematician, Alexander Dewdney.

I've always intended this exploration to be more aesthetic than mathematical.
it is certainly informed by mathematics, but I feel that any great mathematical
insights are unlikely to come from me. I maintain however, the vaguely
plausible fantasy that my unscientific endeavours might inspire some more
useful research by a suitably qualified mathematician at some point in the
future.

==== Problem/Aim ====

The specific aims of this research project are to get some insight into the
structure of the parameter space that these strange attractors occupy. the name
of the project comes from what I hope is a correct name for the full class of
equations which make up the parameter space I am interested in searching.

The specific system of equations which I used in my initial simple program is
shown below:

Xn+1 = a1 + a2Xn +
a3Xn2 + 
a4XnYn +
a5XnZn + a6Yn +
a7Yn2 +
a8YnZn + a9Zn +
a10Zn2

Yn+1 = a11 + a12Xn +
a13Xn2 +
a14XnYn +
a15XnZn + a16Yn +
a17Yn2 +
a18YnZn + a19Zn +
a20Zn2

Zn+1 = a21 + a22Xn +
a23Xn2 +
a24XnYn +
a25XnZn + a26Yn +
a27Yn2 +
a28YnZn + a29Zn +
a30Zn2

The terms a1 though to a30 are the parameters which
define a particular attractor. When I refer to "parameter space", I am refering
collectively to these values. In the same way that you could take two numbers
to be coordinates on a two-dimensional map, it can be useful to imagine a long
list of parameters as coordinates into a space of considerably higher
dimension, in this case, a 30 dimensional space. This is not to suggest that
30-dimensional space is easily comprehensible, but by extrapolation from a two
dimensional plane, through to a three dimensional cube and beyond, it is
possible imagine at least some of the properties of this space.

Perhaps the most important aspect to comprehend is that sets of parameters
where the corresponding values in each set differ only slightly from each other
(eg, a7 in each set) are in some sense adjacent.

The right hand side of equation is simply a polynomial of degree 2 in
Xn, Yn and Zn. As mentioned in the previous
section, it is easy to generalise these equations and replace them with
polynomials of higher or lower degree (greater or fewer numbers of term). With
higher degree polynomials, the number of coefficients (parameters) rapidly
increases.

The level of complexity of the equations can be varied, but even in this simple
(yet still interesting) case the equation is defined by 30 values. If you
consider the 3 starting values of X, Y and Z, then it is defined by 33 values.
When considered as a space, this is a 33 dimensional space.  This raises some
problems of how to sensibly navigate or visualise such high dimensional spaces.

[ automated help looking at the space, fractal dimension, lyapunov ]

It is hoped that the ability to explore and derive a structural understanding
of the number space, may assist in searching for visually or perhaps even
mathematically interesting regions in space.

I expect there to be a tight feedback loop between initial observations and
future explorations. It is very much like sending off a ship into unchartered
waters. if you find an island, you will probably explore the island.

I'm expecting the parameter space to be a kind of meta-attractor, and to be
interesting in similar ways to the attractors themselves. Similar relationships
are already known to exist between better understood fractals. Each point in
the well known Mandelbrot Set fractal corresponds to a particular configuration
of the similarly popular Julia Set fractal in which the solution is bounded
(Sprott p.353). The exact definition of boundedness in this case is beyond the
scope of this report, but it is sufficient to understand that the Mandelbrot
Set can be understood as a map describing the behaviour of the Julia Set over a
range of different parameters. Similarly, I would like to generate a similar
map describing the presence of Strange Attractors across a range of parameters.

==== Methods ====

This exploration is driven by tools and technology. The discovery of fractals
was catalysed by the availability of computers, as they require large numbers
of calculations and their exploration was not feasible without computational
assistance. Even at the time that the Sprott book was written, generating one
of these attractors would take a several seconds, making the animations of my
early exploratory program infeasibly time consuming. The mapping of the
parameter space involves many times more calculations again, making efficient
implementation a reasonably high priority.

In addition to the raw computational challenges, the other important problem to
face was the construction of a suitable interface for conveniently navigating
the high dimensional number spaces.

[ display issues ... when did i switch to soya? .. not documented, just before
getting the dotplot working i guess ]

My early plans were to make a neat, self contained application, displaying the
navigation interface alongside a 3D rendering of the current strange attractor.
An early problem I needed to solve was the problem of integrating a 3D
rendering into that same window as the navigation interface.

The common methods for producing real-time 3D graphics on computers are biased
towards applications in which everything takes place in the 3D render window,
for example, a 3D game, which takes over the complete display of your computer.
I was to discover that relegating that 3D window to a region within another
application window is not always a simple task. The problem was further
complicated the particular 3D library I had hoped to use (OGRE) and the
language in which I had hoped to do the interface programming (Python).

[ summary of the problem? ]

After a number of different trial and error approaches to the problem, I gave
up on the dream of a neat, single-windowed application and fell to the less
problematic approach of having the 3D render and the navigation interface
appear as two distinct windows.

With this initial structural decision made, I moved to implementing the code to
evaluate the equations which would generate the attractor. I decided to use
SciPy, a python library for performing scientific calculations. It emphasises
the use of matrix operations, and so I arranged the evaluation so that the bulk
of the calculation would be done as simple operations on large matrices. This
method was chosen in the interests of efficiency. While it was not possible to
easily test how efficient the matrix operations in SciPy were, it is reasonable
to assume that they are optimised to some degree.

Soon after completing an initial implementation of the evaluation code, I was
somewhat suprised to discover that using functions provided by SciPy,
generating maps of the parameter space would be much easier than I had
imagined. In fact, I was able to render my first parameter maps long before I
had a working renderer for the 3D dot plots of the attractors themselves.

[ randomly chosen attractor 168coeffs.. is that degree 3? ]

The first plots were quite time consuming. For each point on the map I was
calculating many iterations of the equations, only stopping if the values
became incalculably large (infinte). During the ample opportunity for
reflection afforded by the long rendering times, I was reminded of the way in
which the popular images of the Mandelbrot set are typically generated.

Most images of the Mandelbrot set are called "Escape Time" plots. Each pixel in
the image represents a unique starting value which is fed into an iterative
equation. The black region in the centre of the image are those starting
conditions for which the succesive values produced by the iteration stay close
to their initial value, and define the Mandelbrot set proper. The coloured
bands surrounding this region represent starting values for which sucessive
iteration causes the values to spin off towards infinity. The different colours
represent the number of iterations required for the values to cross some
arbitrary threshold.

For my purposes, escape time plots had several benefits. Firstly, there is
generally less calculation to do for each point, as the values will cross the
chosen escape threshold long before my old limit of infinity.
Secondly, while not technically representing attractors themselves, the shape
of the colour bands tend to give clues about the presence of nearby attractors,
acting as a kind of mathematical aura (the allusion to pseudo-science is
particularly appropriate).

[ animations ]

[ optimisations (psyco, liboil, handcoded c) ]

[ dotplot renderer, interface ]

[ basin of attraction 0,0,0 assumption ]

[ sprott mention of robustness ]

==== Solution/Results ====

[ intial plots of basin of attraction (accidentally) .. ]

For historical value, below is the first map I rendered. Mistakenly, it was the
result of somewhat misplaced ambition. It was intended to be a plot of the
varying fractal dimension across a two-dimensional slice of the parameter
space. Due to a conceptual error, it is actually a rendering of the varying
fractal dimension across a two-dimensional slice of the basin of attraction.
Upon realising this error, the slight variance in the values towards the center
became quite surprising.

[ desc of basin of attraction, why the factal dimension shouldn't change much ]

highf-100.png (accidental basin plot)

[ low order ]

[ fractal dimension plot + composite ]

[ basin of attraction animations ]

[ render errors? ]

==== Discussion ====

the features of the parameter space maps are as interesting as i had hoped.

i didn't expect to be making maps this early. i had not anticipated that the manual-searching part of the tool would be so difficult, and i also i had not realised that it would actually be easier to do the mapping. i had expected to move on to the mapping after having an 'explorer' of sorts.

i've yet to speak to anyone with a maths background to ask if this is of any relevance, or more realistically to find out that it is very boring mathematically. i can imagine that it might be considered a little boring because i am working with big complicated polynomials. most of the famous fractals are based on very simple equations.

[ benefits of flexible lib vs app ]

i still want to continue, making a useable application for exploring the space. at the moment i am still driving it quite manually.

[ basin plots are truly 3d (not slices), lend themselves to 3d printing ]

[ ability to click on an map and see the attractor ]

[ sonification ]

[ reactions from DE ]

==== References ====

* http://www.rawilson.net/chaos/lyapunov/index.html

* http://sprott.physics.wisc.edu/sa.htm
  * http://ibiblio.org/e-notes/MSet/Logistic.htm
 
  * images and animations at  [[Pix Strange Attractor]]

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