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> Chaos Issue: 2012-1 Section: University

Greek

 

Chaos! A word we hear so often and scares us so much. Something reasonable enough if one considers that we live in a universe where everything works perfectly and everything can be predicted by certain laws (no matter if they are understandable or not) or at least that is what we think.

Scientifically, Chaos is defined as endmost sensitiveness in initial conditions. In Mathematics and Physics (as well as many other sciences), Chaos Theory, which appeared all of a sudden before forty years, has brought radical changes in the researchers’ way of thought and offered us progress that the classical research methods would never imagine. The present article is attempting an approach of what is Chaos in Physics Sciences, how the necessity for research in chaotic systems was born and in what scientific fields it has applications.

 

The First Steps into Chaos: “The n Body Problem”

Since his beginning, the Man wanted to explain the phenomena taking place all around him. In this attempt of his he began to create models based either on some metaphysical powers or in some natural order that, he suspected, existed in the universe. These models often led to laws that explained, or even predicted, Nature. This was the technique that created “Science”. However, the research results were not always that good.

The first scientist who ever faced a problem not able to be solved was, possibly, the English mathematician and theoretical physicist sir Isaac Newton (1642-1727). As known, Newton created the Law of Universal Gravitation. According to it, any object is falling towards Earth as well as the Earth is moving around the Sun because of a force, known as gravitational. The mathematic description of the force is:

 

 

 

This law manages to explain (mathematically too) why the planets are moving around the Sun in elliptic orbits. However, in comparison to what we believe in school when we are taught this law, Newton didn’t manage to find the motion equation of a body moving under gravity. Meaning, he didn’t manage to find a law that predicts the position and the velocity of a body moving under the gravitational attraction of one or more others in any moment given its mass and the initial position and velocity.

The reason of this gap in the Law of Universal Gravitation is exactly the existence of Chaos. Newton, after solving approximately the system of two bodies moving under gravitation (e.g. the Earth motion around the Sun), considered easy the addition of more bodies. But, with the addition of one more body (for example the Moon in the Sun-Earth system) the system was not solvable. Newton quitted quite early without mentioning anything about this problem in his book “Philosophiae Naturalis Principia Mathematica” (Mathematical Principles of Physics Philosophy) where he expresses his opinions no Mechanics and Gravitation.

During the centuries, many scientists worked on the “three body problem” or, in its general name, the “n body problem”. Everyone thought that with a lot of imagination and a great lot of work, the problem would be solved, as the “two body problem” was. No one was ready to admit that the world we live is not working like a well-tuned clock, as Newton’s laws were ordering. But, Nature is far more complex than we think.

In the late 1980’s, king Oscar II of Sweden (1829-1907) put to scientists from all over the world four problems suggested by German mathematician Karl Weierstrass (1815-1897). Anyone who would manage to solve one of them until the 21st January 1889 (60th birthday of the king) would not only win a great monetary prize (2,500 crowns) but also a golden medal, the publication of his solution and (of course) glory. One of the four problems was the following:

“Given a system of arbitrarily many mass points that attract each according to Newton's law, under the assumption that no two points ever collide, try to find a representation of the coordinates of each point as a series in a variable that is some known function of time and for all of whose values the series converges uniformly.”

Actually, the scientists are asked to solve an n body system: to find the motion equations of n bodies in the three-dimension space under the attraction of gravity and prove that this system is stable (it is not changing despite any disorders). This was the problem the French mathematician, theoretical physicist and science philosopher Henri Poincaré (1854-1912) decided to solve.

Poincaré immediately understood that the problem had to be limited, ending in the three body problem Newton was not able to solve. After that, he studied the orbits of the (three now) planets in the phase space, a space where the coordinates are not given by the position and the time, but by the moment (mass multiplied by velocity) and the position. In this study, Poincaré showed his genius by setting one of the bodies’ mass much less than the others’ (limitation of the problem) and then by studying only a small part of the phase space each time instead of the full orbit.

The second step was abolishing. Poincaré was using a plane right to the orbits of the three bodies (known as Poincaré map) and was noting on it the spot of a body crossing the plane there. In a plane like this is easy to find the periodic and the non-periodic orbits, depending on whether the spots are repeated after some time or not.

With this tool, Poincaré believed he would manage to prove the stability of such a system. And, indeed, after many efforts he did it, deposing to the scientific committee the king Oscar II had chosen a 200-pages solution. The paper had such an impact towards the committee members, that Poincaré was awarded the prize immediately and many thought of suggesting him for the Nobel prize.

However, Swedish mathematician Edvard Phragmén (1863-1937) found a mistake in the solution. Its print in Acta Mathematica journal stopped immediately and Poincaré turned again to the problem. This time the result was the opposite: not only the n body system was not stable but it gave the most paradox solutions the French mathematician had ever seen.

Chaos had made his first timid appear on the scientific stage.

Clouds and Turbulence: Chaos strikes back!

Almost seventy years were needed to pass until the scientific research would turn again to Chaos. One of the pioneers on the matter was the American mathematician and meteorologist Edward Lorenz (1917-2008). Lorentz had created on his computer a “game” about the weather he had built a digital universe whose laws he was able to control and which was simulating (quite realistically) the atmosphere of Earth. Every minute the computer was typing on a page a series of numbers that were describing the weather of one day. The worthy of noticing in all this “game” was that no one of the phenomena was repeated.

In fact, Lorenz set a universe whose weather was evolving according to time. He was using Newton’s laws so that everything works perfectly. But, actually, Lorenz himself was using a form of “ordered disorder” to give his weather the changeable form of the real weather. That way every phenomenon of the weather could be compared with circles repeated again and again without, though, being the same.

The computer, as mentioned, printed on a page a series of numbers describing the weather. Those numbers, for space saving, had three decimals (e.g. 0,506), while the computer made calculations using numbers with six decimals (0,506127). Once, Lorenz wanted to reexamine a long attendance of weather phenomena. Instead of starting the same proceeding from the beginning (insuring he would have all the previous steps with accuracy), he inserted to the computer the numbers he had from the previous time (losing the accuracy of the last three decimals) and began the proceeding from the middle.

The result was a weather that, after some time, had nothing in common with the weather he examined before. The difference in the three last decimals (infinitely small for our approaches) caused with the passing of time a great change in the weather and made Lorenz shout: “A butterfly flying in Brazil can cause a typhoon in Texas!”

After the first shock, Lorenz continued his work in such systems with non-linear equations (equations that express relations with non-severe proportion). He noticed behaviors similar to these in the liquid turbulence. He also studied a certain waterwheel, a system composed by a wheel and small pots hanging on its perimeter. Each pot has a small hole in its bottom and from above the wheel drips water on a stable order. When the water drips slowly the pot that is above all does not fill because of the hole

If the water drips a little quicklier, the weight of the pots is dragging it down making the wheel to turn. In the water drips very quickly, the heaviest pots go down before the empty and the wheel is turning once to one trend and once to the other

Lorenz discovered that things were infinitely more complex.

Using those three equations (with three variables), Lorenz was able to describe fully the motion of the system.

 

 

 

 

 

 

 

 

 

 

Using the triad of numbers resulting from these three equations for a certain moment as coordinates in a three-dimension phase space, Lorenz attempted to make a diagram of the motion he was working on. The result made him famous: a figure infinity complex that looked like a butterfly wings or an owl eyes, the Lorenz Attractor.

Lorenz’s job was continued by the American theoretical biologist Robert May. In the early 1970’s May studied the increase of populations in an ecosystem as a result of the food granting by the area. He noticed that the equation he created showed a certain periodicity that after some time was became unpredictable.

May continued his work on ecosystems and this way came to the result that every system tracing regular periods has the possibility to end up in a disordered (chaotic in other words) evolution. Thanks to his research, the point where a dynamical system is passing to Chaos was accurately determined.

Since then, Chaos Theory has found applications in many sciences. Most characteristic examples are Economy and Astronomy.

In Economy, and mostly in Microeconomics (that studies isolated economic subjects or businesses), Chaos has made a very dynamic entrance. Since 1970’s, many economists and mathematicians are reexamining a lot of economic theories under the prism of the new science. The result was the creation of new econometric techniques able to explain and predict the falling or rising of shares in the markets. For example, the dramatic changes in the Chinese markets the last year are explained through the butterfly effect.

In Astronomy, and specifically in Celestial Mechanics, whose place of studies are the orbits of the celestial bodies (e.g. comets), and in Dynamic Astrophysics, which is interested on the motion and the forces between celestial objects in great body clusters (e.g. planetary systems, galaxies, nebulae), Chaos is the basic research field. It is notable that there is now a new and more probable destruction-of-the-world theory. Because the solar system is, as mentioned, unstable, the planets, especially the small ones (Mercury, Venus, Earth, Mars), are in orbits we cannot predict for more than 100.000.000 years (approximately). Some are suggesting that they will leave their orbit destroying the sensitive balance of the solar system far before the collapse of the Sun (the second most probable destruction scenario).

Maps, Computers and Shares: Fractals everywhere!

Almost the same period, French-American mathematician Benoit Mandelbrot (1924-2010) began to discover a certain proportion by studying accidental and unpredictable alterations (similar to those studied by Lorenz).

Mandelbrot’s interest in Chaos began when he was studying the changes in the price of cotton in several markets (London, New York, etc.). According to the then prevalent theories, the small and temporary changes in the prices had no affection on the big alterations. Everybody’s belief was that small changes happen accidentally, without affecting the total. Mandelbrot, although, dove deeper in the matter studying not only the prices, but the scale they were moving.

After careful research, he realized that, although the changes of the prices were accidental (as all economists were suggesting), the curves of these changes respective to the time were following certain proportions: the curves representing the daily alteration of prices were the same as those representing the monthly or the annual. Mandelbrot had found a sort of order among irregular series of numbers.

That was only the beginning of the revolution that followed. More and more researches lead to the conclusion that Nature is not as regular or Euclidian we think it is, neither is it described by fully logic laws, like those of Newton. Small changes can alter fully a situation that appeared static and immutable. Every system, natural (e.g. the population of a species in an ecosystem) or human-born (e.g. the fluctuation of shares in the markets), shows great complexity. The material itself, in great enlargement, shows abnormalities that have nothing in common with the geometry we know but follow completely Mandelbrot’s ideas.

A new geometry began to take form. Its name: Fractal Geometry. Its figures (Fractals) occupied with their colours and their beauty, the majority of the papers of the corresponding publications. Their main characteristics are the complexity and the self-similarity (stable proportions appeared under enlargement). The most interesting, however, is that the scientists working on this field started to see it everywhere. The Fractal Geometry, they said, is the geometry of Nature.

The Chaos was now the leading star on the stage.

One of the most characteristic examples of the Fractals use is the cartography.

A map (like the map of Great Britain)presents several points (like the coastlines) which do not follow the usual methods of counting lengths and surfaces.

They have infinitely great complexity and the only way of counting the precisely is the continuing division of our measure units: from kilometres to metres, from metres to millimetres. But, if we use a specific Fractal, that has the ability to repeat its figure in every enlargement, the problem becomes clearly easier.

The realistic representation of natural formations, like the three-dimension shape of a mountain, also demands figures not belonging to our familiar (Euclidian) geometry.

So, the computers are using certain Fractals (that era dividing to smaller, similar to the original, figures) in order to give images similar to the real.

Even the nature of matter itself presents great complexity. Enlarging to a smooth surface, we will find out it is not smooth at all. On the contrary, it is rough and possesses a Fractal form, as the molecules are put together disorderly but with a stable scale proportion. The same happens with the snowflakes and the branching out of the nerves.

Today, the Fractals are used even in therapies for the Parkinson disease.

 

Bibliography

  • J. Gleick: “Chaos – Making a New Science”, pp. 34-59, 120-164, Publications “Catoptro” (February 1990), Trans (in Greek): M. Konstantinidis
  • B. Parker: “Chaos in the Cosmos – The stunning complexity of the universe”, pp. 15, 45-52, 75-89, P. Travlos Press (Athens 1999), Trans (in Greek): Th. Grammenos
  • H. Varvoglis: “History and Ideas Evolution in Physics”, pp. 155-159, Thessaloniki Planetarium Publications (Thessaloniki 2011)
  • N.K. Artemiadis: “History of Mathematics (from the mathematician’s point of view)”, pp. 463-465, Academy of Athens – Research Committee (Athens 2000)
  • G. Theodorou: “From Newton to Chaos – Deterministic or Stochastic Description of Nature”, Phenomenon No. 12 (Period 4) (June 2011), pp. 12-15
  • G. Theodorou: “About Chaos”, Phenomenon No. 13 (Period 4) (October 2011), pp.6-9
  • IHMO (In My Humble Opinion) “Chaos Theory – A Brief Introduction” (www.imho.com/grae/chaos/chaos.html), last visited: 27/01/2012
  • J.D. Hadjidemetriou: “Chaos in the Solar System”, Physics Department of Aristotle University of Thessaloniki website(http://users.auth.gr/~hadjidem/solar1.html), last visited: 01/02/2012
  • “? Chaos Guide to Beginners – Chaos and Destruction Theories”, Physics4u Blog(www.physics4u.gr/chaos/chaos6.html), last visited: 30/01/2012
  • B.B. Mandelbrot: “The Fractal Geometry of Nature”, W.H. Freeman Press (San Francisco 1982)
  • H. Varvoglis: “Magic of Fractals”, VimaScience (24/10/2010),(www.tovima.gr/science/article/?aid=362800), last visited: 27/01/2012

 

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Nikolaos Chatzarakis :: 10/10/2012 :: Pageviews: 4877