Brief Introduction to Chaos Theory

Everything in the world has its causes and consequences. Perhaps this thought brought me to the realization that everything in the world is interconnected. There are reasons for everything. Even in chance, there is a movement towards 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” came about due to the fact that the systems described by the theory, taken piece by piece, are unordered, but the Chaos Theory is really to find the hidden order in seemingly random data.

When was Chaos discovered? The first true experimenter in the field of Chaos was meteorologist Edward Lawrence. In 1960, he worked on the problem of predicting the weather. He had a computer installation with a set of 12 equations that simulate the weather (meaning air flows 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 could be.
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Once in 1961, he [Edward Lawrence] again wanted to see a special sequence. To save time, he started from 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 solved differently. Instead of the same model that was before, there was a model deviating very strongly at the end, differing from the original one (see Figure 1). At the end of the ends, he found out what had happened. The computer placed in memory 6 numbers after a comma. To save paper, he entered only 3 numbers after a comma. In the original order, the number was 0.0506127, and it printed only 0.506.

Figure 1 - Lawrence's experiment: the difference in the beginning between these curves is only 0.000127 (Jan Stewart, “Does God Play Dice?”, Chaos Mathematics, p.141)

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

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

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

Starting with the number 2, in the end, you can get a result that is completely different from the results of the same system with the initial number 2.000001. It’s simply impossible to achieve this level of accuracy — just try to measure something to the nearest millionth of an inch! From 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 the initial conditions. His first discovery had 12 equations, and he wanted to simplify it very much, but so that it still had this attribute [sensitivity to difference in initial conditions]. He took the convection equations and made them incredibly simple. This system was no longer related to convection, but was sensitive to differences in the initial conditions, and this time only 3 equations remained. It was later established that these equations describe a whirlpool.

On the surface, water steadily forms a wheel rim. Each “rim” diverges from a small hole. If the water flow has a small speed, the “rims” will never be fast enough for a whirlpool to form. Rotation can continue. Or, if the flow is so fast that the heavy “rims” constantly rotate around the bottom and the surface, the whirlpool can slow down, stop and change the direction of rotation, rotating first in one direction and then in the other. (James Gleick, The Chaos Theory, p. 29)

The equations for this system also seemed to show a general randomness of behavior. Anyway, when the graph was built, he was surprised [Lawrence]. Output parameters always remained on a curve, forming a double helix. Until then, only two types of order were known: a constant state in which variables never change, and a periodic state in which the system is cyclic and repeats indefinitely. 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 obtained equations the Lawrence attractor (see Figure 2).


Figure 2 - Lawrence Attractor

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

Another system in which there is a sensitivity to the difference in the initial conditions is coin tossing. There are two variables in throwing a coin: 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 the coin and how far it flies. It is only possible to place these variables in a specific range, but it is impossible to control them enough to know the result.

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

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

In this equation, the population is described by a number between 1 and 0, where 1 is the maximum possible population and 0 is the extinction. R is an indicator of growth. The question was, how does this parameter affect the population? The obvious answer is that a high population growth rate means 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 we increase the growth rate. At low values, the population was set to a specific value. For an indicator of 2.7, it was set at 0.6292. Further, with an increase in the growth rate of the population “R”, the final population also grew. But then something strange happened.

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

As the parameter rose further, the line bifurcated (forked) again. Bifurcations occurred faster and faster, until they suddenly became chaotic. By setting an accurate growth indicator it is impossible to predict the behavior of the equation. Be that as it may, on closer examination you can see white stripes. Looking closer at these strips, we find a row of small windows, where a line passes through the bifurcations, before returning to a state of chaos. This resemblance to itself is the fact that the graphic is an exact copy of him, hidden deep inside. This has become a very important aspect of chaos. (Figure 3)


Figure 3- Bifurcation

IBM employee Benoit Mandelbrot was a mathematician who studied this self-similarity. One area he studied was cotton price fluctuations. No matter how the cotton price data was analyzed, the results were not distributed normally. Mandelbrot eventually obtained all the available data on cotton prices, up to 1900. When he analyzed the data with the help of a computer, he noticed a striking fact: the number from the point of view of normal sales was symmetrical relative to the point of view in scale. Every single price changed randomly and unpredictably. But the calculation of changes was independent of scale: the curves of daily and monthly price fluctuations absolutely coincided. Amazingly, Mandelbrot's analyzed 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 other phenomena as well. 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 will be missed that are too small to be shown on the map. It is just like when we walk along the shore we miss the microscopic gaps between the grains of sand. It does not matter how much to increase the coastline, there will be more visible intervals when approaching.

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

The approximate Koch curve looks exactly the same as the original. This is another example of self-similarity.



Koch curves contain an interesting paradox. Each time a regular triangle is added, the length of the line becomes larger. But be that as it may, the internal area [limited] of the Koch curve always remains smaller 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. Fractal comes from the word fractional. Fractal fragmentation of the Koch curve is approximately 1.26. Fractal crushing is impossible to think up, but it makes sense. The Koch curve is coarser than a smooth curved line with single crushing. Since it is coarser and more “wrinkled”, it occupies better space. Be that as it may, it is not as good at filling the space as a square with two crushing, since it does not have an area. This means that the fragmentation of the Koch curve is less than 2.

Under the fractal is meant any image that has a self-similarity. The bifurcation diagram of the population equation is a fractal. Attractor Lawrence fractal. 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 the Chaos is a mathematical phenomenon, most of the research in the field of Chaos was done by people who are specialists in other fields, such as meteorology and ecology. Studying the area of ​​spreading Chaos was a hobby for scientists working on a problem, what to do about it.

Later, a scientist by the name of Figenbaum again investigated the diagram of bifurcation. He investigated 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 acquires the property of self-similarity.

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

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

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 it square and then add a number. Enter the result in the square and add a number. Repeat further, and if the number goes to infinity, it is not a Mandelbrot equation.

Fractal structures have been seen in many areas of the real world. The blood is carried through the blood vessels branching on and on, the branches of the tree, the structure of the lungs, the charts of stock sales data, and other systems of the realm of the world have something in common: they all have self-similarity (self-repetition).

Scientists at Santa Cruz University have found Chaos in a tap [how it drips]. Recording the drop of the crane and the periods of time, they opened the exact flow rate, the drops did not fall at the same time. When they plotted the data, they found that drops actually fell with a certain pattern.

The human heart also beats with a chaotic pattern. The time between beats is not constant, it depends on how active a 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 establish heartbeat within certain limits, instead of uncontrolled randomness.

Chaos has application even in science. Computer images become more realistic when applying 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 almost but not exactly repeats itself.

Also with the help of fractals music can be created. Using Lawrence's attractor, Diane 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 coordinates x of Lawrence's attractor, by running the program on a computer, she created variations on the theme of this work. Most of the musicians who heard these new sounds said that the variations are very musical and creative.

UPD: Thanks ixside . "Chaotic Variations on a Classical Theme" are available here. Edit: moved to Popular Science.