Is the Lorenz attractor a fractal

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1. Chaos

A growing number of scientists are now designing new, daring perspectives: "The structure and behavior of living systems in their variability and complexity are as close to chaos as they are to a regular pattern." With the help of the computer, they show us what is only just beginning to be learned, the laws of chaos, irregularities, the unpredictable, which are behind most things in our world (the human heartbeat and human thinking, clouds and thunderstorms, the structures of galaxies, the creation of a poem, the spread of a forest fire, a winding coastline, origin and evolution of life). How order breaks down and merges into chaos, and how chaos creates order, this is where the scientific disciplines overlap. The common tool of all chaos researchers is the computer.

In everyday language, chaos is the opposite of order. Greek = shapeless primal mass. According to Hesiod (700 BC) in the beginning the chaos (the boundless, yawning void) from which Gaia (the earth) arose. In the Bible: "... and the earth was desolate and empty". Ancient peoples believed that the forces of chaos and order were part of an uncomfortable tension. They thought of something immeasurable and creative. The Egyptians imagined the early universe as a shapeless abyss called Nut. Nut gave birth to Ra, the sun. In a Chinese creation story, a ray of pure light, yang (masculine principle), springs from chaos and builds the sky, while the heavy cloud that remains, yin (feminine principle), forms the earth. Too much yin or yang will bring back the chaos. The mythical notion is based on the notion that the cosmic creative power is based on a reciprocal relationship between order and disorder. In the model, order appears as a small area within a wide field of chaos. In chaos there are islands of stability. Mathematicians and natural scientists speak of “chaotic” systems when their development is not determined, not predictable. The term “deterministic chaos” appears paradoxical, which at first glance appears to be mere coincidence, but arises in accordance with strict laws (e.g. in turbulence). Nevertheless, the behavior of deterministically chaotic systems cannot be calculated because they react to the smallest change in the initial conditions. (Geo Knowledge 93)

In the rationalistic-mechanistic worldview, there was no place for chaos in science. Only the “unsolvable” mathematical problems opened the doors and made chaotic processes, paradoxes and turbulences in abundance understandable and visible. Chaotic processes can be triggered by the slightest change. An unstable equilibrium suddenly turns into a large imbalance. The existing order suddenly begins to play “crazy”. Chaos can occur in all areas of life and is not programmable. We can only cope with the fear of chaos by accepting its unpredictability.

1.1 Examples of chaotic phenomena:
  1. Plumes of smoke: First, smoke (cigarette, smoking sticks) rises linearly. But soon it reaches a point where it begins to expand in a whirling manner.
  2. Weather: For a long time it was believed that with enough data from many measuring stations it would be possible to calculate the exchange between high and low pressure areas and thus predict the weather in the longer term. The slightest change in a factor that influences the weather can, however, have unexpected and chaotic effects.
  3. 3. Space: Once upon a time, people still believed in the harmony of the spheres and the idea that everything in space moves along its path in a beautiful, rational way. Today we know Hyperion (a moon of Saturn) who staggers around "chaotically".
  4. Drop formation on the tap: After logical consideration, the water jet would have to become thinner and thinner downwards - but it forms droplets.
  5. Traffic chaos: With the growing traffic, everything doesn't just slow down a bit, but at some point after chaotic fast-paced and slow phases the collapse suddenly comes.
  6. Climate change in South America «El Niño: In South America there is an area in which the climate occasionally changes due to the rise in temperature by several degrees, which leads to consequential damage such as animal deaths.
  7. Pendulum: A pendulum that I hang on a second cord develops completely different chaotic oscillations depending on the initial impulse. The path of a chaotic pendulum can no longer be calculated in advance. When we let go of it, it follows the mechanical laws first. The smallest changes cause something unpredictable (butterfly effect) and the causality is no longer correct.
  8. Eddy formation in the river, mountain stream, canal: Two identical boards flow in the canal with a strong causality (predictable). In the brook with stones, however, no statement is possible.
  9. Fox / mouse model: x1 = ax (1 - x)> Formula with starting point fox population (x1), brake term (1-x) for feedback. What does the development depend on? From the number of starting animals? No: multiplication factor! More foxes> fewer mice, no mice> extinction of the hunters. Chaotic development phenomena with apparently stable intermediate phases, until suddenly the system collapses. What is effective and how can it be influenced?
  10. Wave formation
  11. Sudden forces of nature
  12. Video feedback
  13. Stock exchange
  14. economy
  15. Life, illness, cardiac arrest, death
  16. Teaching and learning

1.2 Nonlinear Equations

In nonlinear equations, changing one variable can have a completely, even catastrophic, effect on other variables. For example, it can be used to describe how your earthquake erupts when two plates on the surface of the earth's crust press against each other. The resulting tension can gradually increase for decades until a “critical” value is suddenly reached. In the non-linear world, however, exact predictions are practically as well as theoretically impossible. However, models can be used to identify critical points in systems where a small change can have a major impact.

1.3 attractor

In simple, regular systems, periodic behavior occurs (periodic oscillations with recurring return to the initial conditions (violin string, pendulum, day and night). The oscillations of a pendulum move from the greatest deflection on the left (zero impulse) to the lowest point (greatest speed) to the largest deflection on the right (impulse zero). Shown in phase space, the pendulum, braked by air resistance, strives towards the point of rest. Mathematicians call this point an attraction point or "attractor". The attractor is an area in phase space that has an attraction force A chaotic attractor has a self-similar, fractal and unpredictable structure and lifespan. The course of its trajectory is sensitive to initial conditions. It can have several attractors at the same time exist. Attractors that appear to be insignificant can bring about unexpected effects in a flash. Attractors in chaos are limited in time, since all factors not involved in the attractor develop further during a stability phase and lead the attractor to an end at some point. Collapse> uncertainty, reorientation, fear of chaos, ...

 

Examples of attractors:

  1. Root extraction: Choose any number and extract the root several times:> always 1. After many iterations (feedback), the number is always aimed at attractor 1.
  2. Chaotic pendulum: A pendulum always swings to the greatest height left and right (there speed 0). But in the end it swings out at the bottom (attractor).

Lorenz attractor:

Lorenz discovered the non-linear dynamic system of weather in 1960. A starting point (state) determined with measurements receives an immeasurable indeterminacy through the connection with the overall system. The true state can be in any point of the attractor. E.g. weather - locally unpredictable, but globally stable.

1.4 turbulence

Turbulence occurs everywhere in nature (air currents, lava, washing around stones, weather disasters) and often pose great problems for people. With evenly flowing water, stable eddies are created that have a point-like attractor and return to the same basic pattern after disturbances. With increasing flow velocity, the eddies become unstable, fray (limit cycle), and finally every water particle (eddy in eddy in eddy) seems to move randomly and chaotically (strange attractor).

1.5 iteration, feedback

From at. «Iterare» = broken. Feedback through resumption and reintegration of everything that was before (e.g. renewal of all body cells in about 7 years, artificial intelligence, weather systems). Repeated application of a calculation rule, each result serving as the starting point for the next step. Negative (inhibiting) and positive (amplifying) feedbacks were discovered in microphones that are too close to the loudspeaker and produce a chaotic noise by sending the captured sound back to the amplifier. Feedback is tension between order and chaos. Feedback occurs everywhere: on all levels of life, in psychological processes, in the evolution of the overall ecological system and in mathematical, non-linear equations.

1.6 Examples of iterations

- Number doubling (exponential growth). x1 = 1, x2 = 2, x3 = 4; 8, 16, 32, x1 = 1.5, x2 = 3, x3 = 6; 12, 24, 48,

- doubling of the numbers with omission of the integer part x1 = 0.9134, 2xn = 1.9134> 0.9134; x1 = 0.707070; 0.414141; 0.828282; 0.656565; 0.313131; 0.626262; 0.252525; 0.505050; 0.010101; 0.020202; 0.040404; 0.080808; 0.161616; 0.323232; 0.646464; 0.292929; 0.585858; 0.707070. After 17 iterations, we end up with the starting number.

- x1 = 0.707070; If we make a small mistake in the 4th decimal place> 0.707170, the error inflates up to the 11th iteration in such a way that the sequence of numbers is completely different from the original. 0.707170; 0.414341; 0.828682; 0.657365; 0.314731; 0.629462; 0.258924; 0.517849; 0.035698; 0.071396; 0.142792; 0.285584

- Not only equations are extremely sensitive to their initial conditions. Researchers observe the same dynamics in liquids. E.g. small eddies in the blood stream, in which brookable points can flow side by side or land in completely different areas of the liquid (butterfly effect).

- The stretching and folding of a baker's dough graphically shows the movements as they occur in non-linear iterations. Adjacent points on the dough come apart, creating a complex, unpredictable pattern.

- Strange attractors and iterations are found in the middle of orders and can cause phenomena such as a heart attack.

1.7 Example of algebraic iteration

An iteration is when a calculation is carried out with a number, then the same calculation is carried out with the result as the starting point, the same calculation with the new result, etc. The iteration and the associated feedback represent one of the most important "tools" in the development of mathematical systems Fractals. The principle of feedback and iteration is very old. It was known to Sumerian mathematicians around 4,000 years ago (for example, they used iterative steps to calculate the square root of a number).

With the calculator and the term square number we can gain a first insight. A real number c is chosen as a constant and a sequence of numbers is now produced.
a0 = 0 Starting with the initial number a0, here 0, an arithmetic operation is started
a1 = a02 + c executed, with the same result, etc. In short: a sequence of
a2 = a12 + c iterations calculated.

In general the sequence tends towards ƒ or a limit value determined by c, which we call g (c). For example, we get g (0.2) ª 0.28 and g (-0.5) ª-0.37. If you enter g (c) into a coordinate system via c, a curve is initially obtained. But if you go to smaller (c), then something surprising occurs. At (c) -1.1, for example, they commute members of the sequence that strive with an even index against a different number than those with an odd index. These numbers are called attractors because they attract the numbers of the sequence, as it were. The curve branches off to the left in the axis cross. A c now has two numbers g (c). In the case of transitions to even smaller c values, it becomes apparent that these two also branch out again. We then get 4 attractors, then 8, Š. At around c '-1.5 a completely new situation arises. Now no structure can be seen at all. One speaks of the structurelessness of chaos. The mathematician Feigenbaum illustrated this in his “fig tree diagram” (see workshop item chaos, bifurcation). We now investigate what small changes in the calculation process (an + 1 = an2 + c) cause if we substitute for a = (0.5 / 1.5 / and C = (1 / -1 /). The result of the second iteration (a = 0.5; C = -1) is surprising. There are two sequences with different limit values, namely -1 and 0. These two numbers are called attractors (more attractive = to use). With the initial number a = 1.5 leads to the same Attractors, but the starting number 2 tends towards ƒ.

 

The discovery that two numbers that differ as little as they want from one another behave completely differently during iteration, i.e. lead to completely different results, can be compared with certain situations in nature. Two closely neighboring raindrops can get into the North Sea or the Mediterranean Sea over a watershed (butterfly effect).

1.8 The geometric iteration

Fractals can also be obtained purely geometrically by starting from a simple point set M0. Several images are now carried out on it at the same time and the images are combined to form a new point set M1. The next step is the same with M1.

1. M0 is stretched centrically by 0 in a ratio of 1: ÷ 2 and rotated by 0 by 45 °.
2. M0 is stretched in the same way, but now rotated by 0 by 135 ° and shifted by 1 to the right.

1.9 Bifurcation

If the birth rate of any animal population is below 1, the whole population will drop to 0 and go out. But if it is larger, it will first drop, in order then to level off at a constant value (approx. 2/3 of the original size). It seems that 66% has become an attractor. When the birth rate is increased to the critical value of 3.0, something new happens. The attractor 0.66 becomes unstable and splits into two, i.e. the population no longer approaches one value, but fluctuates back and forth between two stable values. A further increase above 3.4495 results in a new split (4 values) and from 3.56 onwards we have bifurcation in eight fixed points (period doubling). From 3.56999 the number of attractors becomes infinitely large and ends in chaos.


2. fractals

From Latin "frangere, fractum" = to break, broken. With the help of fractal geometry, principles of order can be shown in chaos. The term was coined in 1975 by the mathematician Benoit Mandelbrot. Fractals are often self-similar structures, the edges of which are not smooth, but infinitely rough. Each enlargement shows new, similar structures (apple males). The appearance in detail is retained in ever smaller scales. Fractal structures have a broken dimension (cube = 3 dimensions). Many natural forms and non-linear systems have fractal properties or behave fractally (clouds, mountains, bronchi, plants, galaxies, coastlines, lungs, weather, blood circulation, brains). The attraction of the fractals is presumably that there is an image of the whole in each of its “parts”.

2.1 Examples of fractals

  • Z2 + C = any number. The computer sends this formula on a journey into the Mandelbrot crowd. Z is a complex number that can change and C is a fixed complex number. The computer must substitute the result of the addition for Z in the next round.
  • According to Mandelbrot, the fractal structure of turbulence shows that it occurs in gusts. On a stormy night the wind will suddenly subside, then come back up again, let leaves circle until they sink again. This turbulence repeats itself on a smaller and smaller scale.
  • Even the behavioral patterns of the weather, which Lorenz recognized as chaotic, are now considered to be fractal.
  • The ramifications of a living tree are obviously fractal. branches have twigs, these have smaller twigs again, and the details are repeated down to the smallest twig. Leonardo da Vinci already stated that the branches become so thinner as they branch out that the overall thickness (all branches packed together) remains the same.
  • Fractal eight to 30-fold branching structure of the veins and arteries for the blood supply. Fractal self-similarity pervades the bodies of organisms. The body is a network of nothing but self-similar systems.
  • Art: Labyrinths, iterative language games, song patterns, ingeniously interwoven Celtic drawings, archetypal patterns on ritual, old vessels often evoke the feeling of recognition (of what has already been seen) and is related to fractals and attractors. Art also explores the tension between order and chaos.

2.2 Self-similarity in fractal structures

As early as 1900, the mathematician Koch found a fractal structure by dividing a given distance into thirds and erecting an equilateral triangle over the middle third. With the resulting four equally long stretches he now did the same as with the initial stretch, etc. If you start from an equilateral triangle, a snowflake-like structure (snowflake curve) is created which, although closed, is infinitely long.

The resulting shapes are self-similar. A figure is called self-similar when parts of the figure are small copies of the whole figure. Ferns, for example, as a whole, in the struts and the sub-struts, also show self-similarity. A fern leaf drawn on the screen with a fractal program can hardly be distinguished from a real one. Other examples from nature: cauliflower, the branches of a tree, ...

“You will risk losing your childish ideas about clouds, galaxies, leaves, feathers, flowers, rocks, mountains, torrents, carpets, stones and many other things. There will be no going back to their old conception of these things. " (Michael F. Barnsley)


2.3 How long is the coast of Great Britain?

We have similar phenomena when we want to calculate the length of a coast and choose the fineness down to the size of a grain of sand. If you measure the length with rods in meter size, you could say that the number of rods I need to lay them out along the coast corresponds to the size of the coast of Great Britain. But if you did the same thing again with shorter sticks, you could follow finer curves of the coastline. If this length were added, the result would be a larger coastline. The smaller the rods, the finer the branches that they can follow. The length of the coast grows to infinity. These structures, like the said coast, are called fractals and their length is given in the fractal dimension. This number is unimaginable, because it does not appear in the usual view of the world. There are no three dimensions, but fractal structures have a fraction (= fractus) as the number of their dimension. Seen from a great distance, the coast consists of several large bays - the closer you focus on the coast, the more smaller bays open up in the large bays.

2.4 nonlinear fractals (Mandelbrot and Julia sets)

2.4.1 Amount of almond bread

The most famous representatives of this group of fractals are undoubtedly the Mandelbrot set (also called the M set) and the Julia sets (Gaston Julia, 1893-1978). These quantities have caused quite a stir since Mandelbrot was introduced at the end of the 1970s and thus made available to the public. They are the birthplaces of the most famous and beautiful fractal images in the world. To this day, they have inspired many artists, kept asking scientists new questions and magically attracted the public with their fascinating splendor of colors and graceful variety of shapes.

The Mandelbrot set is the classic fractal. It's just a graph in the complex plane of numbers. The x-axis represents the real part, the y-axis the iminary part of a complex number. There are no isolated points in the complex number plane: the Mandelbrot set is connected. The top and bottom halves of the images are axially symmetrical to each other. The main motifs seem to repeat (but never exactly the same) when enlarged. The order in chaos: fractals resemble each other again and again themselves. The set of all complex numbers c, for which the sequence does not strive against ƒ, form the so-called almond bread quantity (apple male).

The mathematician Mandelbrot extended Feigenbaum's investigations to the set of complex numbers and thus opened the door to the world of fractals. A point of the complex number plane corresponds to a c. Mandelbrot colored this point black if and only if the associated iteration sequence does not strive towards the amount against ƒ. The amount of black dots created in this way is called the amount of Mandelbrot or, because of its shape, the “apple male”. If you hike along the edge of the apparent curve, you have to constantly change your course even in the smallest sections, which is why Mandelbrot speaks of a fractal based on the word break (frangere, fractus).

«Clouds are not spheres, mountains are not cones, coastlines are not circles. The bark is not smooth - and even lightning does not make its way straight ... The existence of such forms challenges us to study what Euclid leaves aside as formless, leads us to the morphology of the amorphous. So far, however, mathematicians have avoided this challenge. By developing theories that are no longer related to visible things, they have moved away from nature. In response to this, we will develop a new geometry of nature and demonstrate its usefulness in various fields. This new geometry describes many of the irregular and fragmented shapes around us - with a family of figures we shall call fractals. " (Benoit Mandelbrot 1975 - The fractal geometry of nature)

Since the discovery of fractal geometry almost 20 years ago, opinions have been divided about its importance, be it in mathematics or in other areas. Some stamped fractals as irrelevant colorful pictures that represent just a freak of nature and are not worth further consideration or investigation. The others, on the other hand, were overflowing with euphoria and promised themselves revolutionary new insights that would completely change the previous worldview from the new world, into which they penetrated full of overzealousness and which they explored with an almost childish enthusiasm and fascination.

 

2.4.2 Julia sets

The Julia sets are named after the French mathematician Gaston Julia. The Julia set uses the same iteration rule as the Mandelbrot set, namely zn + 1 = zn2 + c, only here the complex number c always remains constant. For this, the starting value z0 is now the interesting thing.