A Brief Introduction to Chaos Theory

Original author: grae (http://www.imho.com/grae)
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Everything in the world has its own causes and consequences. Perhaps this thought led me to the realization that everything in the world is interconnected. There are reasons for everything. Even in chance, there is a movement toward a goal.

Events that seem random occur in a certain sequence.
"Even in chaos, there is order."

What exactly is chaos? The name Chaos Theory comes from the fact that the systems described by the theory, taken in pieces, are disordered, but the Chaos Theory actually consists in finding a hidden order in seemingly random data.

When was Chaos discovered? The first true Chaos experimenter was meteorologist Edward Lawrence. In 1960, he worked on the problem of weather prediction. He had a computer installation with a set of 12 equations simulating the weather (meaning air flow in the atmosphere) [clarification here]. They themselves did not predict the weather. But be that as it may, the computer program theoretically predicted what the weather might be.

Once in 1961, he [Edward Lawrence] again wanted to see a special sequence. To save time, he started in the middle of the sequence, instead of doing it first. He entered the numbers from the printout and launched the program ...

When he returned an hour later, the pattern was resolved differently. Instead of the same model as before, there was a model that deviated very strongly at the end, differing from the original (see Figure 1). In the end, he found out what happened. The computer placed 6 decimal places in the memory. To save paper, he introduced only 3 decimal places. In the original order was the number 0.0506127, and he printed only 0.506.
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Figure 1 - Lawrence's experiment: the difference at the beginning between these curves is only 0.000127 (Ian Stewart, “Does God Play Dice?”, Chaos Mathematics, p. 141)

According to the generally accepted opinion of that time, this should have worked. He was supposed to get an order very close to the original. The scientist could consider himself lucky, having received measurements with an accuracy of 3 decimal places. Of course, it was impossible to measure the 4th and 5th digit using rational methods, and this could not affect the result of the experiment. Lawrence thought the idea was wrong. This effect is known as the Butterfly Effect. The difference in the starting points of the two curves is so small that it is comparable to the fluttering of the wings of a butterfly [in real life].

The movement of the wings of one butterfly today creates the slightest change in the state of the atmosphere. As time passes, the atmosphere is different from what it could be. Thus, after a month, a tornado that could fall on Indonesia does not appear. Or, if he was not to appear, he appears. (Ian Stewart, “Does God Play Dice?”, Mathematics of Chaos, p. 141).

This phenomenon, commonly referred to as Chaos Theory, is also known as a sensitive dependence on initial conditions. Just a small change in the initial conditions can dramatically change the behavior of the system considered for a long period of time. Such a small difference in measurements can be caused in the experiment by noise, background noise, or equipment malfunction. These things cannot be avoided even in the most isolated laboratory.

Starting from number 2, in the end, you can get a result that is completely different from the results of the same system with an initial figure of 2.000001. It's just impossible — to achieve that level of accuracy — just try to measure something to the nearest millionth of an inch! Based on this idea, Lawrence found it impossible to accurately predict the weather. Be that as it may, this discovery led Lawrence to other aspects of what later became known as Chaos Theory.

Lawrence began to observe the simplest systems that are sensitive to differences in initial conditions. His first discovery had 12 equations, and he wanted to simplify it very much, but for it to have this attribute [sensitivity to difference in initial conditions]. He took convection equations and made them incredibly simple. This system was no longer related to convection, but it was sensitive to the difference in the initial conditions, and this time only 3 equations remained. It was later found that these equations describe a whirlpool.

On the surface, water is steadily forming a kind of wheel rim. Each “rim” diverges from a small hole. If the water flow has a low speed, the “rims” will never become fast enough to form a whirlpool. The rotation may continue. Or, if the flow is so fast that the heavy “rims” revolve all the time around the bottom and surface, the whirlpool can slow down, stop and change the direction of rotation, turning first in one direction and then in the other. (James Gleick, Chaos Theory, p. 29)

The equations for this system also seemed to show general randomness of behavior. Be that as it may, when the graph was built, he was surprised [Lawrence]. The output parameters always remained on the curve, forming a double helix. Prior to this, only two types of order were known: a constant state in which variables never change, and a periodic state in which the system is cyclical and indefinitely repeated. Lawrence's equations were definitely ordered — they always followed in a spiral. They never stopped at one point, but never repeated the same state, that is, they were not periodic. He called the resulting equations the Lawrence attractor (see Figure 2).

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Figure 2 - Lawrence Attractor

In 1963, Lawrence published an article describing his discovery. He included an article on weather unpredictability and discussed all types of equations that caused this type of behavior. Unfortunately, the only journal in which he could publish his article was the meteorological journal, since he was not a physicist or mathematician, but a meteorologist. As a result, Lawrence's discoveries were not known until they were discovered again by other people. Lawrence discovered something revolutionary, and waited for someone to open it.

Another system in which there is sensitivity to the difference in the initial conditions is coin flipping. There are two variables to coin tossing: how soon it will fall and how fast it spins. Theoretically, it is possible to control these two variables completely, and to control how the coin falls. In practice, it is impossible to control exactly the speed of rotation of a coin and how much it flies up. It is only possible to place these variables in a certain range, but it is impossible to control them enough to know the result.

A similar problem occurs in the ecology and prediction of biological populations. The equation is simple if the population grows definitely, but predators and limited food make this equation incorrect. The simplest equation is:

next year's population = r * this year's population * (1 - this year's population) [where next year's population is the population next year, this year's population is the population this year]

In this equation, the population is described by a number between 1 and 0, where 1 represents the maximum possible population, and 0 is extinction. R is an indicator of growth. The question was, how does this parameter affect the population? The obvious answer is a high rate of population growth means setting a high level, while a low one means that the population will fall. This condition is true for some growth indicators, but not for all.

Biologist Robert May, decided to find out what will happen to the equation, if you increase the growth rate. At low values, the population was established at any particular value. For the indicator equal to 2.7, it was set at 0.6292. Further, with an increase in the growth rate of the “R” population, the total population also grew. But then something strange happened.

As soon as the indicator exceeded 3, the line was divided in two. Instead of setting in any particular position, she “jumped” between two different values. It had one meaning in one year, and completely different in the next. And so this cycle was repeated constantly. An increase in the growth rate caused jumps between two different values.

As soon as the parameter increased further, the line bifurcated (bifurcated) again. Bifurcations occurred faster and faster, until suddenly they became chaotic. By establishing an accurate growth rate, it is not possible to predict the behavior of the equation. Be that as it may, with a closer examination you can see white stripes. Looking at these strips closer, we find a series of small windows where a line passes through the bifurcations before returning to the state of chaos again. This resemblance to oneself is the fact that the graphic is a replica of himself hidden deep inside. This has become a very important aspect of chaos. (Figure 3)

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Figure 3- Bifurcation

IBM employee Benoit Mandelbrot was a mathematician who studied this self-similarity. One of the areas he studied was the fluctuation of cotton prices. No matter how the cotton price data was analyzed, the results were not distributed normally. Mandelbrot ultimately received all the available data on cotton prices, right up to 1900. When he analyzed the data using a computer, he noticed a striking fact: the number in terms of normal sales was symmetrical relative to the point of view in scale. Each individual price changed randomly and unpredictably. But the calculation of the changes was independent of the scale: the curves of daily and monthly fluctuations in prices absolutely coincided. Surprisingly, Mandelbrot’s price changes remained constant throughout the noisy period of the 60s, World War II and the depression.(James Gleick, Chaos - Making a New Science, p. 86)

Mandelbrot analyzed not only cotton prices, but also other phenomena. One of them was the length of the coastline. A map of the coast shows many bays. But be that as it may, when calculating the length of the coastline, small bays that are too small to be shown on the map will be missed. This is similar to how, when walking along the shore, we miss the microscopic gaps between the grains of sand. No matter how much the coastline increases, there will be more visible gaps when approaching.

One mathematician, Helge von Koch, took this idea for mathematical construction called the Koch curve. To create a Koch curve, imagine an equilateral triangle. Draw another equilateral triangle towards the middle of each side. Continue to add new triangles to the midpoints of each side, and the result will be a Koch curve (see Figure 4).

Koch's approximate curve looks exactly like the original. This is another example of self-similarity.

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Koch curves contain an interesting paradox. Each time another triangle is added, the length of the line becomes longer. But be that as it may, the internal area [limited] of the Koch curve always remains less than the area of ​​the circumscribed circle around the first triangle. That is, it is a line of unlimited length, enclosed in a limited area.

To understand this, mathematicians used the concept of fractal. The fractal comes from the word fractional. The fractal fragmentation of the Koch curve is approximately 1.26. Fractal fragmentation cannot be invented, but it makes sense. The Koch curve is rougher than a smooth curve line, which has a single fragmentation. Since it is coarser and more “wrinkled”, it takes up space better. Be that as it may, it is not as good at filling the space as a square with two crushing, because it has no area. This means that the fragmentation of the Koch curve is less than 2.

By a fractal we mean any image that has self-similarity. The bifurcation diagram of the population equation is fractal. The Lawrence attractor is a fractal. The Koch curve is also a fractal.

At this time, scientists found it difficult to publish works on Chaos. Since they have not yet shown his attitude to the real world. Most scientists did not think that the results of experiments on Chaos are important. As a result, even though Chaos is a mathematical phenomenon, most of the research in Chaos was done by people who are specialists in other fields, such as meteorology and ecology. Studying the spread of Chaos was a hobby for scientists working on the problem of what to do about it.

Later, a scientist by the name of Figenbaum again examined the bifurcation diagram. He examined the rate of onset of bifurcation. He discovered that it comes at a constant rate. He calculated that this number is 4.669. in other words, he determined the exact scale at which the bifurcation curve becomes self-similar.

Reduced by 4.669 times, the diagram looks like a subsequent bifurcation region. He decided to look at other equations to see if it is possible to apply the scale factor to them as well. To my great surprise, the scale factor turned out to be the same. Not only for complex equations describing a regularity. The regularity was exactly the same as for simple equations. He tried many functions and they gave a scaling factor of 4.669.

This was a revolutionary discovery. He discovered a whole class of mathematical functions that behave identically, predictably. The versatility has helped many scientists easily analyze the equations of chaos. She gave scientists the first tools for analyzing chaotic systems. Now they could use simple equations to get the result of more complex ones.

Many scientists have discovered equations that create fractal equations. The most famous fractal image is the simplest. It is known as the Mandelbrot equation. The equation is simple: z = z 2+ c. To find out if your equation is such, take the complex number z. Get its square and then add the number. Enter the result into the square and add the number. Repeat further, and if the number tends to infinity, this is not the Mandelbrot equation.

Fractal structures have been seen in many areas of the real world. Blood spreads through blood vessels branching further and further, tree branches, lung structure, stock sale data charts, and other systems of the real world have something in common: they all have self-similarity (self-repetition).

Scientists at the Santa Cruz University have found manifestations of Chaos in a water tap [the way it drips]. By recording the drops falling from the tap and the time periods, they discovered the exact flow rate, the drops did not fall at the same time. When they plotted the data, they found that the drops actually fall with a certain pattern.

The human heart also beats with a chaotic pattern. The time between strokes is variable, it depends on how active the person is at the moment, and on many other things. Under constant conditions, the heartbeat can still accelerate. Under various conditions, the heart beats uncontrollably. This can be called a chaotic heartbeat. Heartbeat tests can help medical research find a way to set the heartbeat within a specific framework, instead of uncontrolled randomness.

Chaos has applications even in science. Computer images become more realistic with Chaos and fractals. Now, using a simple formula, you can create a beautiful, realistic-looking tree on your computer. Instead of following a normal pattern, tree branches can be created using a formula that repeats itself almost, but not exactly.

Also, using fractals, music can be created. Using the Lawrence attractor, Diana S. Debbie, a graduate in electronic engineering from the Massachusetts Institute of Technology, created musical themes. (“Bach to Chaos: Chaotic Variations on a Classical Theme”, Science News, Dec. 24, 1994). By associating the musical notes of a piece of music from the Bach Prelude in C with the x-coordinates of the Lawrence attractor, by running the program on a computer, she created variations on the theme of this work. Most musicians who heard these new sounds said that the variations are very musical and creative.

UPD: Thanks ixside . "Chaotic Variations on a Classical Theme" is available here. Amendment: moved to the Popular Science.

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