deterministic chaos, fractals and the mandelbrot set - a simple introduction

8/7/2019 Deterministic Chaos, Fractals and the Mandelbrot Set - A Simple Introduction

1/6

Deterministic Chaos, Fractals and theMandelbrot Set A Simple IntroductionBy Matthew Mitchell

Deterministic chaos differs from complex systems, which are systems which produceemergent behaviour too complex to be modelled by linear equations. Such systems areresistant to reductionism since they are a sum of their parts. This is a feature of many projectsinvestigating A-life, but also of economies and markets.

Except where noted, the following sections are based on Kaye (1993)

Determinism itself is an old idea. Many theories of Newton (1642 - 1727) were based ondeterministic processes.

Laplace (1749 - 1827) also applied determinism to his studies of the universe in his book "Celestial Mechanics".

Theories of determinism were upset in 1927 by Hiesenberg's (1901 - 1976) UncertaintyPrinciple , which states it is impossible to measure both position and velocity with completecertainty. Einstein was one scientist who was reluctant to abandon determinism.

Deterministic chaos applies to systems which are theoretically completely predictable, buttheir sensitivity to initial conditions makes practical prediction impossible.

In the following we will look at deterministic systems called Fractals (from fractured). Inparticular , we will look at some of the more well known work related to fractals called The

Mandelbrot set (all of the following material is based on Kaye (1993), which is an excellent,accessible and interesting book on this topic).

Cellular Worlds Again

The Mandelbrot Set is based on discrete pixelised worlds that divide space up kind of like achequer board.

The axes are marked off in parts as follows:


2/6

This means that the pixel location is (1.0, 1.0), and the pixels are of size 0.1.

In Fractals pixel size is important, because we can dive into greater levels of detail, For example, we can now expand our pixel above to look at pixels of size 0.01 (i.e increase our magnification by an order of magnitude):

We can do a similar magnification process with 1D cellular models:


3/6

The numbers in the top boxes indicate how many times the address of the box must besquared before it exceeds a maximum value (in the case of addresses > 1.0) or drops below aminimum value (in the case of addresses < 1.0).

The formula for this is as follows:

n + 1 = n2 [n]

Where n is a large number. This means we square the address value, then square that valueagain up to n times.

The values (addresses) greater than 1 will approach infinity, while those less than 1 willapproach zero.

Since there is relatively small number of values in the boxes, we could demonstrate the

amount of squaring a number was subjected to before it exceeded (or dropped below) athreshold value, by replacing the numbers with an arbitrary colour scheme as follows:

Note, that in this example the change between numbers which approach infinity and thosewhich approach zero is very sudden (i.e either side of 1.0).

R epresenting Points

The Mandelbrot set is a process like we saw above, where numbers are multiplied bythemselves and we decide whether they approach infinite or zero. In the simplest case, if theyapproach infinity we colour them black, if they approach zero they are white. Later on weconsider how we can add additional colours to our scheme based on how quickly the numbersapproach their asymptotic values (infinity or zero).


4/6

The numbers we are using are based on pixel address in 2D space (although the Mandelbrotset can be seen as 3 dimensional).

Points we are used to

Number representations that everyone are already familiar with in 2D spaces are x and y coordinates in a Cartesian plane (even if you can't remember this very well).

The x and y representations of points ( x, y), can be combined to form a line typically in anequation of the form:

y = m x + b

Here m is the slope of the line, x is the x intercept, and b is a constant.

In a Cartesian coordinate system, given a point (x,y), x is measured on one dimension, and y is measured on another, at 90 degrees, if we were to include a third coordinate ( z) it would be

at a third 90 degree angle.

An Alternative R epresentation of Points

An alternative form of locating points in 2D space is to use the following notation:

P1 = (a + ib)

a = x co-ordinateb = y coordinatei = direction of measurement of b at right angles to a (a direction may be up, down or left or right)

In this form, the number (point) is said to consist of two parts:

a = the real partib = the imaginary part

The P in this case is called a complex number and is represented by c. The a is equivalent to x and the b is equivalent to y. A complex number is a bit strange in that it has this third part, thei.

The

i in imaginary

When dealing with complex numbers i is the square root of -1. This number plays a role inchanging the x and y (a and b) values when complex numbers are multiplied (remember partof what we are interested in the squaring of numbers).

While the square root of -1 cannot be evaluated, because of the way complex numbers aremultiplied and the fact we square complex numbers means we sometimes obtain: ( the squ ar e


5/6

roo t of -1)2 which can be substituted with -1. It is this which is used to effect the movementof the point as we apply the squaring operation.

T he movement of complex numbers

A complex number is always composed of two parts, even once squared. The a and b components can always be plotted on a 2D grid (in practice this gets difficult if they approachinfinity.)

The following figure shows hypothetically how a point represented by a complex number (such as: 0.25, 0.35) may move as the square operation is applied repeatedly 9 times.

In each of these iterations we apply the following update rule:

n + 1 = n2 + c [n]

n is initially 0, c is the complex number, c ie. the point based on the address of a pixel, inthis case (0.25, 0.35).

The resulting number n + 1 can always be divided into two parts, the real a and imaginary ib parts.

Note, the multiplication used for the squaring is a bit unusual. This is omitted from our discussion.

Attractors for Complex Numbers

Earlier we were interested in determining whether a number, when squared, would approacheither infinity or zero. These were the attractors for these numbers.

The attractors for complex numbers is either infinity , in which case the a and b componentsboth approach infinity, or some other number, much less than infinity. In the diagram above,the attractor after 9 iterations appears to be the point (0.2, 0.4).


6/6

The problem is that is can take a long time (i.e many iterations) to try and determine whether a complex number is going to shoot off to infinity or not, as they tend to bounce around(behave erratically) a lot on the way.

T he Black and White Mandelbrot Set

In the simplest scheme, numbers which go to infinity are considered outside the Mandelbrotset and are coloured white, while numbers approaching some other attractor are consideredtrapped by an attractor (other than infinity) and are counted as being inside the Mandelbrotset and are coloured black.

Behaviour of Points Near the Boundary

In our 1D example earlier we saw that there was a sharp boundary between numbers whichapproached zero and those which approached infinity.

Points in the Mandelbrot set near the border of the set tend to take a long time to settle on an

attractor or go to infinity. We can help resolve this by magnifying the cells on the border. Bydiving into higher resolution we can examine the cells at a higher resolution and distinguishthose on either side of the border. Again those right on the border in this new resolution, canbe further magnified, and this process can continue infinitely.

Adding more Colours to the Set

The cells that are not clearly in or out of the set (at a particular magnification) could becoloured differently based on how quickly they move away from the origin of the 2D space.Distance from the origin over time (iterations) tends to be a good indicator of whether a pointwill stay in the set or escape.

Those that stay in the set can remain coloured black, while those that escape (or appear toescape) at different speeds can be given different colours based on their velocity. This leadsto the colours seen in the Mandelbrot set.

Some points outside of the main set may also approach attractors other than infinity, so wesometimes get black points which are disconnected from other cells in the Mandelbrot set.These of course can be magnified to higher resolution, and the same familiar patterns willemerge.

Here are some Mandelbrot images

deterministic chaos, fractals and the mandelbrot set - a simple introduction

Documents