### Comments about Wikipedia: Chaos Theory

Following is a discussion From Wikipedia, the free encyclopedia
• The text in italics is copied from that url
• Immediate followed by some comments
In the last paragraph I explain my own opinion.

### Chaos theory

In mathematics, chaos theory describes the behavior of certain dynamical systems – that is, systems whose state evolves with time – that may exhibit dynamics that are highly sensitive to initial conditions (popularly referred to as the butterfly effect)
• The problem with the sentence is that its starts with the word "in Mathematics", this gives the impression that we are going to study the Chaos Theory from a theoretical point of view i.e. by studying certain diferential equations. You could. But you should state clearly that this is your intention.
• It is much more practical to replace the words "In Mathematics" by "The". That means you are going to study certain physical system in mathematical language.
• The question becomes than which are those dynamical system.
• The butterfly effect is in principle a physical effect.
As a result of this sensitivity, which manifests itself as an exponential growth of perturbations in the initial conditions, the behavior of chaotic systems appears to be random.
• If the intention here again is only in mathematics, than it that means we compare two set of initial conditions. It is correct that the difference between the two outcomes can exponentially grow.
• On the other hand if you study a physical system this exponential grow is much more difficult to demonstrate or to prove.
This happens even though these systems are deterministic, meaning that their future dynamics are fully defined by their initial conditions, with no random elements involved.
• In mathematical sense this is correct, assuming that the system is mathematical deterministic. SEE "Deterministic system (mathematics)"
• In reality how do you know that a physical system is determistic? Are all physical systems deterministic? If that answer is "yes", than the word Determistic has no meaning.
• See also "Deterministic system (philosophy) " The meaning is very tricky. In fact in that document they make a difference between: (1) Some deterministic systems (2) Non-deterministic systems and (3) Systems with controversial classification.
• Do we call the solar system deterministic starting from a gas cloud 4 billion years ago until the present with a Sun, planets and an earth occupied with human beings which have a mind ?
This behavior is known as deterministic chaos, or simply chaos.
• Which behaviour ? Which of the above do they mean ? I expect the fact that certain systems are highly sensitive to initial conditions.
Chaotic behaviour is also observed in natural systems, such as the weather. This may be explained by a chaos-theoretical analysis of a mathematical model of such a system, embodying the laws of physics that are relevant for the natural system.
• Here we clearly mean a physical system. But is this deterministic or non deterministic or in between ?
• How is chaotic behaviour observed ? Bij monitoring rain or by looking to the clouds ?
• The important question is what is the mathematical model that describes the weather.
• IMO it is very difficult to find an accurate model, and that is the main reason why weather prediction is so difficult.
• The same is true for the climate.

### Overview

Skip
Systems that exhibit mathematical chaos are deterministic and thus orderly in some sense; this technical use of the word chaos is at odds with common parlance, which suggests complete disorder.
• In the first half mathematical (theoretical) chaos is discussed, in the second half physical chaos. Of course this can easily leads to conflicts.

### Reflection

There are two mayor problems with the above text
1. First the text is not clear. The main reason is a mathematical explanation versus a physical explanation. Many text books high light the mathematical approach, that means there main points are around the differential equations involved, and they minimize the physical difficulties.
2. The second reason is what clearly is the chaos theory. Is this only the fact that certain differential equations are highly sensitive to initial conditions ? This is true in mathematical sense but much more difficult to demonstrate in physical sense i.e. related to the real world.
3. A third reason is the usage of the word deterministic. Deterministic systems imply order. That means are predictable. Non Deterministic implies less order, less predictable or more unpredictable. In that sense is the word deterministic chaos very misleading.
4. Pierre Simon Laplace (See the book "Pierre Simon Laplace 1749 - 1827 A life In exact Science" by Charles Coulston Gillispie) and others, used two concepts in relation to deterministic, namely order and cause and effect. With the first he expressed that there is stability in the solar system. With the second change in time, the relation between past versus future expressed in mathematical laws.
The question is how exact can we describe the physical world in mathematical laws.

The following link also discusses "deterministic chaos" Here we read:
A system is chaotic if its trajectory through state space is sensitively dependent on the initial conditions, that is, if unobservably small causes can produce large effects

In the last few decades, physicists have become aware that even the systems studied by classical mechanics can behave in an intrinsically unpredictable manner. Although such a system may be perfectly deterministic in principle, its behavior is completely unpredictable in practice. This phenomenon was called deterministic chaos.

• Here we read that "certain systems are deterministic in principle". What does that mean. ?
• How can a physical system be deterministic in principle ? but not in practice. ?
• It seems strange that if a system is completely unpredictable to call that deterministic in principle.

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Created: 5 june 2008

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