Entry

Math: Chaos theory: What is the relationship between fractals and chaos?

Nov 29th, 2003 10:10
Knud van Eeden,

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--- Knud van Eeden --- 29 November 2003 - 06:04 pm -------------------

Math: Chaos theory: What is the relationship between fractals and 
chaos?

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Chaos becomes of particular interest when you investigate dynamic 
systems.
That are for example systems that change with the time.

Typical examples are the

 -weather

 -mechanical systems like 

  pendulums
  (single pendulum, double pendulum, ...),

  waterwheel (which is a rotating wheel
  that can carry water from a lower to a higher level,
  like used by the ancients)

 -planetary systems like the movement of three bodies (say the earth
  and mars revolving around the sun as an example of this).

If you want to investigate this, your efforts boil down to and can be
compressed to a set of equations.

Because you investigate changes here, you often see that these
equations will be differential equations.

Once Laplace said basically that given the equations and enough
information we should be able to predict exactly what was going to
happen.

This turned out not to be the case, at least in the mathematical
universe (which mimics the physical universe via these equations).

As soon as you have a system of coupled first order non-linear
differential equations with at least three independent variables,
mathematical chaos might occur.

Thus any system of coupled differential equations in the form, 
starting from
this:

 f' = f(x,y,z)

 g' = g(x,y,z)

 h' = h(x,y,z)

This means that if you change the values of the parameters in this
equations just a tiny little bit, very large changes might result in
some cases.

So a small change in the input gives a large change in the output
(before the name 'chaos' arrived on the scene, they might have called
this an instabile system).

Henri Poincar<e'>, a French mathematical physicist, around the
beginning of the century jumped upon this chaotic behaviour, when he
amongst others was trying to win the big price to be paid by the king
of Sweden for anybody who was able to predict if our planetary system
was stabile or not.

In that case you might quite naturally start to investigate the
behaviour of 2, 3, 4, 5, 6, ... (planetary) bodies, under influence of
gravity (so governed by Newton's second law which says force equals
mass times acceleration).

This leads to systems of coupled differential equations, which will
show this chaotic behaviour.

One relationship between fractals and chaos occurs, say, when you want
to plot the behaviour of this dynamical systems.

In the case of a nice behaving pendulum (say you are sitting on a
swing, and move up and down perpetually), if you plot the energy versus
your speed, this might be represented as a nice symmetric ellipse.

As this figure is basically some different occurence of a line, the
dimension of this figure is going to be one-dimensional.

In the case of a chaotic behaving pendulum (say you are sitting on a
swing, which hangs underneath another swing, or thus a double
pendulum), if you plot the energy versus your speed, or even its
position in the air) this might be represented as a very complicated
looking figure with a lot of curls and swings.

As this figure is basically some different occurence of a lot of lines,
potentially able to fill the whole area, or thus two dimensions, the
dimension of this figure is shown to be between one-dimensional and two
dimensional. A dimension between two whole numbers, say dimension 1.8
is called a fractal dimension, because it is the answer to the question
'what is the dimension of a fractal figure' (like the Koch snowflake).

So one of the relationships is that if you plot certain aspects of the
behaviour of such dynamical systems, figures might occur, which have a
non-integer dimension. And non-integer dimensions are typically
involved when you ask for the dimension of fractal figures.

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Book: see also:

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[book: see also: author: Gleick, James - title: chaos - publisher: 
Sphere books - year: 1988 - pages total: 352 - ISBN: 0-7474-0413-5 - 
position: p. 42 'Chaos: Revolution' - URL: 
http://www.amazon.com/exec/obidos/ASIN/0140092501/qid=1025897733/sr=2-
1/ref=sr_2_1/103-1173699-4890212#product-details]

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Internet: see also:

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Math: Chaos theory: Overview: Can you give me an overview of chaos 
theory?
http://www.faqts.com/knowledge_base/view.phtml/aid/26929/fid/867

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