Susse Chandrasekaran

CHAOS (kā'ŏs') Noun

A lack of order or regular arrangement: clutter, confusedness, confusion, derangement, disarrangement, disarray, disorder, disorderedness, disorderliness, disorganization, jumble, mess, mix-up, muddle, muss, scramble, topsy-turviness, tumble. - Websters

The term "CHAOS" really has nothing with whether or not a system is frenzied or wild . In fact, a chaotic system can actually evolve in a way which appears smooth and ordered

"A chaotic system is defined as one that shows sensitivity to initial conditions. That is, any uncertainty in the initial state of the given system, no matter how small, will lead to rapidly growing errors in any effort to predict the future behavior…In other words, the system is chaotic. Its behavior can be predicted only if the initial conditions are known to an infinite degree of accuracy, which is impossible." 

- Gollub and Solomon

"A chaotic system is defined as one that shows sensitivity to initial conditions. That is, any uncertainty in the initial state of the given system, no matter how small, will lead to rapidly growing errors in any effort to predict the future behavior…In other words, the system is chaotic. Its behavior can be predicted only if the initial conditions are known to an infinite degree of accuracy, which is impossible." 

- Gollub and Solomon

The first true experimenter in chaos was a meteorologist, named Edward Lorenz. In 1960, he was working on the problem of weather prediction. He had a computer set up, with a set of twelve equations to model the weather

One day in 1961, he wanted to see a particular sequence again. To save time, he started in the middle of the sequence, instead of the beginning. He entered the number off his printout and left to let it run.

When he came back an hour later, the sequence had evolved differently. Instead of the same pattern as before, it diverged from the pattern, ending up wildly different from the original

In the original sequence, the number was .506127, and he had only typed the first three digits .506

This effect came to be known as the Butterfly Effect. The amount of difference in the starting points of the two curves is so small that it is comparable to a butterfly flapping its wings. The flapping of a single butterfly's wing today produces a tiny change in the state of the atmosphere. Over a period of time, what the atmosphere actually does diverges from what it would have done. So, in a month's time, a tornado that would have devastated the Indonesian coast doesn't happen. Or maybe one that wasn't going to happen, does.

This phenomenon, common to chaos theory, is also known as SENSITIVE DEPENDENCE ON INITIAL CONDITIONS

At the top, water drips steadily into containers hanging on the wheel's rim. Each container drips steadily from a small hole. If the stream of water is slow, the top containers never fill fast enough to overcome friction, but if the stream is faster, the weight starts to turn the wheel. The rotation might become continuous. Or if the stream is so fast that the heavy containers swing all the way around the bottom and up the other side, the wheel might then slow, stop, and reverse its rotation, turning first one way and then the other.

The equations for this system also seemed to give rise to entirely random behavior. However, when he graphed it, a surprising thing happened. The output always stayed on a curve, a double spiral. There were only two kinds of order previously known: a steady state, in which the variables never change, and periodic behavior, in which the system goes into a loop, repeating itself indefinitely. Lorenz's equations were definitely ordered - they always followed a spiral. They never settled down to a single point, but since they never repeated the same thing, they weren't periodic either. He called the image he got when he graphed the equations

THE LORENZ ATTRACTOR.

Benoit Mandelbrot believed that fractals were found nearly everywhere in nature, at places such as coastlines, mountains, clouds, aggregates, and galaxy clusters. He currently works at IBM's Watson Research Center and is a professor at Yale University. He has been awarded the Barnard Medal for Meritorious Service to Science, the Franklin Medal, the Alexander von Humboldt Prize, the Nevada Medal, and the

Steinmetz Medal for his works

One of the areas he was studying was cotton price fluctuations. No matter how the data on cotton prices was analyzed, the results did not fit the normal distribution. Mandelbrot eventually obtained all of the available data on cotton prices, dating back to 1900. When he analyzed the data with IBM's computers, he noticed an astonishing fact…………………….

The numbers that produced aberrations from the point of view of normal distribution produced symmetry from the point of view of scaling. Each particular price change was random and unpredictable. But the sequence of changes was independent on scale: curves for daily price changes and monthly price changes matched perfectly. Incredibly, analyzed Mandelbrot's way, the degree of variation had remained constant over a tumultuous sixty-year period that saw two World Wars and a depression.

Helge von Koch, captured this idea in a construction called the Koch curve. To create a Koch curve, imagine an equilateral triangle. To the middle third of each side, add another equilateral triangle.

The Koch curve brings up an interesting paradox. Each time new triangles are added to the figure, the length of the line gets longer. However, the inner area of the Koch curve remains less than the area of a circle drawn around the original triangle. Essentially, it is a line of infinite length surrounding a finite area.

Keep on adding new triangles to the middle part of each side, and the result is a Koch curve A magnification of the Koch curve looks exactly the same as the original. It is another self-similar figure.

Cantor took a line segment of length x.

He decided then it would be fun to take out the middle third as his first step in building his little object.

Then, he divided the remaining segments by three as well, taking out the middle thirds of those.

Then he repeated it once more

. . . . . . . . . . . . . . . And so on till…

Follwing this to ad infinitum two startling facts became apparent:

1.If one kept doing that forever, he wouldn't end up with a finite set of zero-dimensional. There would be an infinite number of them, and they weren't really points but bits of a line segment cut to lengths of infinite smallness. 2.This random collection of points was not a one-dimensional line either, as we'd begun by cutting it up, and infinite iteration of that process would never yield anything capable of having length (a one-dimensional measurement).

So in a lower dimension Cantor's Dust (as it came to be known) was infinite when measured, but in the next possible step up it had a one-dimensional measure of zero.

Lorentz ,Mandelbort ,Cantor and Koch had made one thing painfully clear

The convenctional method of measurement i.e whole dimesions cannot measure everything

Fractal dimensia is not just a part science, but one that helps Everyman to see the same world differently. Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line Benoit Mandelbrot

Fractal comes from the word fractional. A fractional dimension is impossible to conceive, but it does make sense. The Koch curve is rougher than a smooth curve or line, which has one dimension. Since it is rougher and more crinkly, it is better at taking up space. However, it's not as good at filling up space as a square with two dimensions is, since it doesn't really have any area. So it makes sense that the dimension of the Koch curve is 1.26 somewhere in between the two.

Fractal dimension: hidden dimensions

• Mandelbrot called not intact dimensions – fractal dimensions (for example 2.76)

• Euclid geometry claims that space is straight and flat.

• Object which has 3 dimensions correctly is impossible

• Examples: Great Britain coastline, human body

Fractal has come to mean any image that displays the attribute of self-similarity. The bifurcation diagram of the population equation is fractal. The Lorenz Attractor is fractal. The Koch curve is fractal.

A scientist by the name of Feigenbaum was looking at the bifurcation diagram again. He was looking at how fast the bifurcations come. He discovered that they come at a constant rate. He calculated it as 4.669. In other words, he discovered the exact scale at which it was self-similar. Make the diagram 4.669 times smaller, and it looks like the next region of bifurcations. He decided to look at other equations to see if it was possible to determine a scaling factor for them as well. Much to his surprise, the scaling factor was exactly the same. Not only was this complicated equation displaying regularity, the regularity was exactly the same as a much simpler equation. He tried many other functions, and they all produced the same scaling factor, 4.669.

The Mandelbrot fractal set is the simplest nonlinear function, as it is defined recursively as f(x)=x^(2+c). After plugging f(x) into x several times, the set is equal to all of the expressions that are generated. The plots below are a time series of the set, meaning that they are the plots for a specific c. They help to demonstrate the theory of chaos, as when c is -1.1, -1.3, and -1.38 it can be expressed as a normal, mathematical function, whereas for c = -1.9 you can't. In other words, when c is -1.1, -1.3, and -1.38 the function is deterministic, whereas when c = -1.9 the function is chaotic.

Time Series for c = -1.1

Time Series for c = -1.3

Time Series for c = -1.38

Time Series for c = -1.9

The most famous fractal image is also one of the most simple.The

equation is simple: z=z2+c. To see if a point is part of the Mandelbrot set, just take a complex number z. Square it, then add the original number. Square the result, then add the original number. Repeat that ad infinitum, and if the number keeps on going up to infinity, it is not part of the Mandelbrot set. If it stays down below a certain level, it is part of the Mandelbrot set. The Mandelbrot set is the innermost section of the picture, and each different shade of gray represents how far out that particular point is. One interesting feature of the Mandelbrot set is that the circular humps match up to the bifurcation graph. The Mandelbrot fractal has the same self-similarity seen in the other equations. In fact, zooming in deep enough on a Mandelbrot fractal will eventually reveal an exact replica of the Mandelbrot set, perfect in every detail.

EYECANDY

Tree simulation using Brownian motion and fractal called Pythagor Tree

• Order of leaves and branches is complicated and random, BUT can be emulated by short program of 12 rows.

• Firstly, we need to generate Pythagor Tree.

Tree simulation using Brownian motion and fractal called Pythagor Tree

• On this stage Brownian motion is not used.

• Now, every section is the centre of symmetry

• Instead of lines are rectangles.

• But it still looks like artificial

Tree simulation using Brownian motion and fractal called Pythagor Tree

• Now Brownian motion is used to make randomization

• Numbers are rounded-up to 2 rank instead of 39

Tree simulation using Brownian motion and fractal called Pythagor Tree

• Rounded-up to 7 rank

• Now it looks like logarithmic spiral.

Tree simulation using Brownian motion and fractal called Pythagor Tree

• To avoid spiral we use Brownian motion twice to the left and only once to the right

• Now numbers are rounded-up to 24 rank

Fractals and world around• Branching, leaves on trees, veins in hand,

curving river, stock exchange – all these things are fractals.

• Programmers and IT specialists go crazy with fractals. Because, in spite of its beauty and complexity, they can be generated with easy formulas.

• Discovery of fractals was discovery of new art aesthetics, science and math, and also revolution in humans world perception.

What are fractals in reality?

• Fractal – geometric figure definite part of which is repeating changing its size => principle of self-similarity.

• There are a lot of types of fractals• Not just complicated figures generated by

computers.• Almost everything which seems to be casual could

be fractal, even cloud or little molecule of oxygen.

How chaos is chaotic?• Fractals – part of chaos theory.• Chaotic behaviour, so they seem disorderly and

casual.• A lot of aspects of self-similarity inside fractal.• Aim of studying fractals and chaos – to predict

regularity in systems, which might be absolutely chaotic.

• All world around is fractal-like

Geometry of 21st century• Pioneer, father of fractals was Franco-American professor

Benoit B. Mandelbrot.• 1960 “Fractal geometry of nature”• Purpose was to analyze not smooth and broken forms.• Mandelbrot used word “fractal”, that meant factionalism of

these forms• Now Mandelbrot, Clifford A. Pickover, James Gleick, H.O.

Peitgen are trying to enlarge area of fractal geometry, so it can be used practical all over the world, from prediction of costs on stock exchange to new discoveries in theoretical physics.

Practical usage of fractals and chaos• Computer systems (Fractal archivation, picture compressing without

pixelization)• Liquid mechanics

– Modulating of turbulent stream– Modulating of tongues of flame– Porous material has fractal structure

• Telecommunications (antennas have fractal form)• Surface physics (for description of surface curvature)• Medicine

– Biosensor interaction– Heart beating

• Biology (description of population model)

Susse Chandrasekaran

"Bach to Chaos: Chaotic Variations on a Classical Theme", Science News, Dec. 24, 1994, pg. 428.

Gleick, James, Chaos - Making a New Science, Penguin Books Ltd, Harmondsworth, Middlesex, 1987.

Lowrie, Peter, personal interview over the Internet, May 17, 1995.

Rae, Kevin, "Chaos", unpublished paper, submitted to Professor Gould, Modern Physics class, Claremont McKenna College, December 5, 1994.

Stewart, Ian, Does God Play Dice? The Mathematics of Chaos, Penguin Books Ltd, Harmondsworth, Middlesex, 1989.