AN INTRODUCTION TO CHAOS THEORY

**WHAT IS CHAOS THEORY?**

Chaos | Chapter 1: Motion and determinism

Chaos theory as a name comes from the fact that the systems the theory describes (non-linear systems) would seem to be disordered or at least unpredictable. Chaos theory tries to find some underlying order in what appears to be random events or data.

Edward Lorenz was an early pioneer of the theory. He was working on weather predictions in 1961 and was using a computer to help with the calculations. Lorenz had initiated a sequence of data based on twelve variables in his attempt to predict weather. He wanted to see the sequence again, so re-entered the data. The weather patterns the computer predicted from the new simulation was very different from what had been predicted initially. Working backward, Lorenz discovered that he had entered the data only out to the third decimal point, whereas in the initial simulation, he had used the same data to the fifth decimal point. These differences are really very, very small and, according to the thinking of the day, should have had only a tiny impact, if any, on the resultant predictions.

Exactly how important is all this? Well, in the case of weather systems, it's very important. Weather is the total behaviour of all the molecules that make up earth's atmosphere. And in Quantum Mechanics it had been established that a tiny particle cannot be accurately pin-pointed, due to the Uncertainty Principle. The Uncertainty Principle prohibits accuracy. Therefore, the initial situation of a complex system cannot be accurately determined, and the evolution of a complex system can therefore not be accurately predicted.

This is the sole reason why weather forecasts begin to be inaccurate around more than 5 days into the future. We can't get an accurate fix on the present situation, only a mere approximation, and so our ideas about the weather are doomed to fall into misalignment in a matter of hours, and days. Nature will not let herself be predicted.

Scientists of Lorenz’s time had focused their thinking on linear systems, where the whole is essentially the sum of the parts. However, there had been a number of instances where linear functions did not explain the behaviour of the system, such as with Lorenz. Non-linear systems are much harder to describe since the mathematical equations cannot be added together to produce new systems as with linear systems.

Chaos Theory is a mathematical sub-discipline that studies complex systems. Examples of these complex systems that Chaos Theory helped fathom are Earth's weather system, the behaviour of water boiling on a stove, migratory patterns of birds, or the spread of vegetation across a continent. Chaos is everywhere, from nature's most intimate considerations to art of any kind. Chaos-based graphics show up all the time, wherever flocks of little space ships sweep across the movie screen in highly complex ways, or awesome landscapes adorn the theatre of some dramatic Oscar scene. (These movie landscapes are produced using fractals on a computer.)

Complex systems are systems that contain so much motion (so many elements that move) that computers are required to calculate all the various possibilities. That is why Chaos Theory could not have emerged before the second half of the 20th century.

But there is another reason that Chaos Theory was born so recently, and that is the Quantum Mechanical Revolution and how it ended the deterministic era. Up to the Quantum Mechanical Revolution people believed that things were directly caused by other things, that what went up had to come down, and that if only we could catch and tag every particle in the universe we could predict events from then on. Entire governments and systems of belief were (and, sadly, are still) founded on these beliefs. Chaos Theory however taught us that nature most often works in patterns, which are caused by the sum of many tiny pulses.

**Attractors**

An Introduction to Chaos Theory with the Lorenz Attractor

The first Chaos Theorists began to discover that complex systems often seem to run through some kind of cycle, even though situations are rarely exactly duplicated and repeated. Plotting many systems in simple graphs revealed that often there seems to be some kind of situation that the system tries to achieve, an equilibrium of some sort. For instance: imagine a city of 10,000 people. In order to accommodate these people, the city will spawn one supermarket, two swimming pools, a library and three churches. And for argument's sake we will assume that this setup pleases everybody and an equilibrium is achieved. But then an ice cream plant is built on the outskirts of the town, opening jobs for 10,000 more people. The town expands rapidly to accommodate 20,000 people; one supermarket is added, two swimming pools, one library and three churches and the equilibrium is maintained. That equilibrium is called an attractor.

Now imagine that instead of adding 10,000 people to the original 10,000, 3,000 people move away from the city and 7,000 remain. The bosses of the supermarket chain calculate that a supermarket can only exist when it has 8,000 regular customers. So after a while they shut the store down and the people of the city are left without groceries. Demand rises and some other company decides to build a supermarket, hoping that a new supermarket will attract new people. And it does. But many were already in the process of moving and a new supermarket will not change their plans.

The company keeps the store running for a year and then comes to the conclusion that there are not enough customers and shut it down again. People move away. Demand rises. Someone else opens a supermarket. People move in but not enough. Store closes again. And so on.

This awful situation is also some kind of equilibrium, but a dynamic one. A dynamic kind-of-equilibrium is called a Strange Attractor. The difference between an Attractor and a Strange Attractor is that an Attractor represents a state to which a system finally settles, while a Strange Attractor represents some kind of trajectory upon which a system runs from situation to situation without ever settling down.

The discovery of Attractors was exciting and explained a lot, but the most awesome phenomenon Chaos Theory discovered was a crazy little thing called Self-similarity. Unveiling Self-Similarity allowed people a glimpse of the intriguing mechanisms that shape our world, and perhaps even ourselves...

**Summary: Chaos Theory; An Introduction**

• A tiny difference in initial parameters will result in the completely different behaviour of a complex system.

• The Uncertainty Principle prohibits accuracy. Therefore, the initial situation of a complex system cannot be accurately determined, and the evolution of a complex system can therefore not be accurately predicted.

• Complex systems often seek to settle in one specific situation. This situation may be static (Attractor) or dynamic (Strange Attractor).

**Self-Similarity**

How does nature direct molecules into snowflakes, or crystals or any other regular form? Chaos Theory has an answer: Self-similarity, a fundamental principle that allows building blocks to mimic their own shape in the building they make.

A large number of elements may form a shape that is derived from the shape of one element. And because no element can be coerced to follow a certain path, no large number of elements will display the exact same pattern as another group. Patterns caused by large numbers of elements are alike but never the same. Hence all snowflakes look alike but no two are exactly identical.

Self-similarity is a really important. It occurs all over nature and many have argued that self-similarity is one of the key natural principles that shape our world the way it is. Self-similarity has been observed in all fields of research: physics but also biology and even psychology and sociology.

Self-similarity is the repetition of a shape, form or behaviour on different levels of complexity. Not as an identical copy, but as a variation of the same basic shape.

An image that displays self-similarity is usually called a fractal. One of the first works of Fractal Art was made in 1904 by a Swedish mathematician named Helge von Koch, and his piece was the so-called Koch Snow Flake. He took a triangle and added a similar but smaller triangle to each of the sides of the first one. Then he added smaller triangles to the sides of the second ones, again to the sides of the third ones, and so on at infinitum.

Every textbook reporting on the Koch Snow Flake will demand that if indeed we continue this process at infinitum we will add length to the outline at infinitum, hence producing an infinitely long line. And yes, theoretically this is true, but with every step the added triangle gets smaller and smaller, and in the real world there is no such thing as infinitely small. After a great many steps the sides of the smallest triangle will be one quantum long, and no smaller triangle can be added.

Koch's tiniest triangles are identical to the big triangle, but way down the line this concoction proves to be an impossible structure. It's an un-fractal since the tiniest triangles will lose their form and fuzz up. A sleek triangle like that will only occur in the minds of Koch and perhaps Plato, but not in nature. Nature produces snowflakes that are never the same because the large-scale phenomenon doesn't mimic the shape of the small-scale phenomenon, but the behaviour: unpredictability, randomness and sovereignty. And Math has no way of generating randomness.

**Summary: Self-Similarity**

• Self-similarity is a structure repeated on a different level of complexity or at a different scale.

• Numbers cannot fully represent reality because (1) they are too accurate, and (2) they can't mimic the randomness that comes from the freedom which is the most fundamental principle of nature.

Chaos Theory deals with nonlinear things that are effectively impossible to predict or control, like turbulence, weather, the stock market, our brain states, and so on. These phenomena are often described by fractal mathematics, which captures the infinite complexity of nature. Many natural objects exhibit fractal properties, including landscapes, clouds, trees, organs, rivers etc., and many of the systems in which we live exhibit complex, chaotic behaviour. Recognizing the chaotic, fractal nature of our world can give us new insight, and power. For example, by understanding the complex, chaotic dynamics of the atmosphere, a balloon pilot can “steer” a balloon to a desired location. By understanding that our ecosystems, our social systems, and our economic systems are interconnected, we can hope to avoid actions which may end up being detrimental to our long-term well-being.

**Principles of Chaos**

• The Butterfly Effect: This effect grants the power to cause a hurricane in China to a butterfly flapping its wings in New Mexico. It may take a very long time, but the connection is real. If the butterfly had not flapped its wings at just the right point in space/time, the hurricane would not have happened. A more rigorous way to express this is that small changes in the initial conditions lead to drastic changes in the results. Our lives are an ongoing demonstration of this principle. Who knows what the long-term effects of teaching millions of kids about chaos and fractals will be?

• Unpredictability: Because we can never know all the initial conditions of a complex system in sufficient (i.e. perfect) detail, we cannot hope to predict the ultimate fate of a complex system. Even slight errors in measuring the state of a system will be amplified dramatically, rendering any prediction useless. Since it is impossible to measure the effects of all the butterflies (etc.) in the World, accurate long-range weather prediction will always remain impossible.

• Chaos is not simply disorder. Chaos explores the transitions between order and disorder, which often occur in surprising ways.

• Mixing: Turbulence ensures that two adjacent points in a complex system will eventually end up in very different positions after some time has elapsed. Examples: Two neighbouring water molecules may end up in different parts of the ocean or even in different oceans. A group of helium balloons that launch together will eventually land in drastically different places. Mixing is thorough because turbulence occurs at all scales. It is also nonlinear: fluids cannot be unmixed.

• Feedback: Systems often become chaotic when there is feedback present. A good example is the behaviour of the stock market. As the value of a stock rises or falls, people are inclined to buy or sell that stock. This in turn further affects the price of the stock, causing it to rise or fall chaotically.

• Fractals: A fractal is a never-ending pattern. Fractals are infinitely complex patterns that are self-similar across different scales. They are created by repeating a simple process over and over in an ongoing feedback loop. Driven by recursion, fractals are images of dynamic systems – the pictures of Chaos. Geometrically, they exist in between our familiar dimensions. Fractal patterns are extremely familiar, since nature is full of fractals. For instance: the shape of trees, rivers, coastlines, mountains, clouds, seashells, hurricanes, etc.

**Mandelbrot Set Zoom**

The Mandelbrot set is the set of complex numbers 'c' for which the sequence ( c , c² + c , (c²+c)² + c , ((c²+c)²+c)² + c , (((c²+c)²+c)²+c)² + c , ...) does not approach infinity.

Mandelbrot set images are made by sampling complex numbers and determining for each whether the result tends towards infinity when a particular mathematical operation is iterated on it. Treating the real and imaginary parts of each number as image coordinates, pixels are coloured according to how rapidly the sequence diverges, if at all.

Images of the Mandelbrot set display an elaborate boundary that reveals progressively ever-finer recursive detail at increasing magnifications. The "style" of this repeating detail depends on the region of the set being examined. The set's boundary also incorporates smaller versions of the main shape, so the fractal property of self-similarity applies to the entire set, and not just to its parts.

**Benoit Mandelbrot: Fractals and the art of roughness**

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