Beauty and Unpredictability From Chaos

Can a butterfly flapping its wings change the fate of a city? Why is it difficult to predict the weather? Why living things behave unpredictably? Why is the stock market risky? Why is there so much biodiversity? Can solar system orbits be predicted?

In fact, it all started with the weather forecast. Edward Lorenz was a mathematical meteorologist. Lorenz thought that successful weather forecasts could be made for the long term using data such as temperature, pressure, wind speed, and powerful computers.

To make this prediction, he created a very simplified atmosphere model based on 12 nonlinear equations. This model had to be as simple as possible, since computers in 1995 were even weaker than today’s clock chips.

Lorenz gave the necessary information to the model and allowed the computer to perform a 6-hour simulation. One day he decided to repeat the same simulation twice, same result expected from deterministic equations, but it did not,initially the graphics were similar, then they started to be very different. Lorenz thought the computer was working incorrectly.But then he saw that the problem was caused by different numbers given at the beginning.He entered the number that should have been 0.506127 as 0.506.

A very small difference caused dramatic variations in the unexpectedly progressing process, which can be explained by adherence to the initial conditions. Small differences in initial conditions can make the weather sunny or stormy.

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Laplace’s demon

We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.

Laplace is a French astronomer and mathematician, he mentions about determinism in this quote.An object obeys the laws of physics, if you have
information of position and momentum, you can calculate its future.
For example, it is possible to calculate which trajectory of the thrown ball or where it will fall.If we apply the same principle to every object in the universe, we can fully predict the future (in theory).

Today, we can know that the universe is not deterministic with the theories of chaos and quantum mechanics.

Sensitive Dependence On Initial Conditions


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Chaos appears in many parts of life, we just don’t notice or say luck.
The experiment here is designed to show how sensitive chaotic systems are to initial conditions.

The 2-pendulum is a nonlinear deterministic system, that is, its equations are clear and fit Newtonian mechanics. It is incredibly dependent on the initial conditions that make the system chaotic.All 3 pendulums are released with an angle difference of 0.1 degrees between them we can ask what the change can be so small that even the human eye cannot notice.Such a small difference, as we see in the weather forecast, can cause huge changes as time goes on.

The normal pendulum is a partially non-chaotic system, the system becomes chaotic as it gets more complex and the number of parts increases.

  • It has been realized that systems with almost identical initial conditions can evolve in different directions very quickly.
  • A small error in the measurement of initial values ​​in a linear system progresses as a result of calculations, causing a small error.On the other hand, a small error in the nonlinear system can lead to an incredible error.
  • The linear system is approximately equal to the parts that make up itself, whereas in the nonlinear system it is equal to more or less of the parts that make up it.
  • For example, let’s examine the non-linear 2x²-1 equation. Let’s choose 2 very close starting points 1.005 and 0.995. Using the result we found in the next step, we can see how the results we achieved when we proceeded only 7 iterations differed by 180146 and 0.971.
  • Factors that make the world and the universe work; simple and nonlinear laws are dependence on initial conditions and feedback.
  • No accurate measurement can be made at any time.
    You measured the speed of an object as 3.7 m / s, maybe the actual value is 3.70000000000000005. There may be an infinite number after the point, even a billionth digit error will turn into a snowball effect as time progresses in chaotic systems and it will really distort you.
  • While very sensitive systems can even be affected by the gravitational force of a raindrop 3 km ahead. Can the butterfly’s wing flutter change the temperature or direction of the air in a million trigger a storm?

From chaos to butterfly

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Lorenz çekicisi

In her article, Lorenz asks the question of whether a butterfly’s flapping wings in the Amazon Forest can cause a storm rupture in the USA?
The butterfly metaphor comes from the geometrical structure of the Lorenz attractor resembling a butterfly.Some equations produce patterns that accumulate around certain points instead of random numbers, which are called attractors.

When Lorenz used new values ​​in some of the equations he used, he saw that the graph spiraled and never cut each other, if he would be periodic and predictable. The system is not stable and if no line crosses each other when it goes on forever, there must be infinite lines.There may be infinite different spirals in the limited area of ​​the Lorenz hammer. For example, there is an infinite number between 1 and 0. It is an issue where chaos intersects with fractals.

Biodiversity and possible lives


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Today, the theory of evolution has very simple mechanisms, such as natural selection, these mechanisms cannot explain some events and it is not known exactly how the evolutionary process works.The theories we have established about evolution remain very simple, the evolutionary process is more complex and chaotic than our estimates, especially in macro evolution.

Biological evolutionary processes and dynamics of thought processes are at the peak of complex systems.If an important series of mutations happened differently in the past, would living things be like today?We know that small changes in the starting point can lead to big results in the future.

If this difference is copied for millions of years, we will most likely reach different creatures from today, we must wait millions of years in the alternative universe to know exactly what living things will look like.The result is not exactly predictable. Sensitivity to initial conditions may have caused biodiversity in the world.

Similarly, if there was a slight difference in the past trajectory of the asteroid that hit Earth 65 million years ago, perhaps there would be dinosaurs on Earth right now.

Chaotic behavior seems inevitable because life is very complex. The growth of living populations generally exhibits chaotic behavior.

Feedback and Risky stock market


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In order to be successful in the stock market, it is necessary to make the right investments and forecasts. Stock market is a complex system affected by human psychology, natural and social events. Feedback systems can be seen in complex systems.

The stock market is not always chaotic, but it is often difficult to make exact predictions. Positive and negative feedback may occur in trading processes.For example, if a company is getting valued, people will try to get a share from here, and the value of the company will gradually increase, the appraisal of the company will support people who want to get a share, and the shares will increase the value of the company and will continue to increase in this way ( up to some extend).

When Newton cried

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3-body problem

After Newton published the Principia Mathematica, people thought they had discovered the laws of the universe and could calculate every move perfectly.But there was a problem, although Newton’s laws can fully calculate the orbits of two objects (such as the Earth around the Sun), the same laws cannot calculate the common gravitational force of three or more objects.

This is called a 3-body problem, it can be defined for objects number larger than 2. Equations describing such systems can be written on paper, but they cannot be solved. Approximate solutions can be found with some iterative methods.

If you make difficult calculations with computers that cover all the planets and their effects,the results will be very close to the original orbits of the planets for a short time.These calculations remain within acceptable error limits throughout human life, but it is almost impossible to calculate the trajectories after 1 million years.

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Predicting the future, changing the past

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Why do people not pray for sunrise or predictable lunar eclipses when they pray for rain?

Although the universe is mostly chaotic, transitions from chaos to complexity and order can be seen at special points. The order here depends on the energy coming from the outside. The order on Earth depends on the Sun.But order is temporary, the second law of thermodynamics will someday put out all the stars.

The development of scientific understanding generally brings with it philosophical implications. Even if the universe is completely deterministic and all the future is hidden now, there is no way to know the future, but to watch the evolution of the universe. The universe cannot know its future and is the fastest simulator itself.

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Ömer Özgür

Ömer Özgür


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