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Chaos as we generally understand it has nothing
to do with the mathematical concept of **chaos**.

A typical example of chaos in a seemingly deterministic system (a system where cause and effect but not arbitrariness play a role) is developments on the stock exchange, changes the weather, a sudden traffic jam or the behaviour of quantum particles in physics.

Mathematical chaos describes complex dynamic systems. These time dependent systems often behave quite sensitively and visibly chaotically to the degree that they are unforeseeable; this though while the individual components of the system are subject to explicit and natural laws.

Since the 1970s, mathematics began to study
chaotic systems. Non-linear equations serve to make computations in such
systems. An important process when making such calculations is the fact that
certain computational procedures continually repeat themselves, where the
attained value is inserted into the original value in the same equation and the
result of the equation used once again… Something similar to if we had to
extract the root of 16, then 4, and then once again 2… This repeated use of the
result as the original value in a certain computational procedure is referred to
as **iteration. **

The physicist Mitchell Feigenbaum discovered
definite similarities between the patterns and structures of systems inclined
towards chaos and derived from it certain quantities which he called **
Feigenbaum’s constants** :

If we take a look at Feigenbaum’s diagram, we
can recognise definite and continually repeating types of tree branches, which
we refer to as **bifurcation** and which can be calculated with a computer.

There are three characteristics of chaotic systems:

Sensitivity

Chaotic systems react very sensitively to original values. In short, one can say that the waving of a butterfly’s wings is sufficient to fundamentally change a dynamic weather system. Small initial causes can have extreme effects.

Periodicity

A dynamic system shows similar behaviour in defined time periods. An example of this could be rings of a cigarette smoke.

Mixing

A dynamic system is not convergent and its iteration apparently fulfils an entire set of values. Such a system does not show any periodicity (refer above).

The graphical depiction of complex structures
shows the same forms in detail as is evident on a larger scale. This
characteristic is referred to as **self-similarity** and an example of which
is the individual heads of a cauliflower or the branches of a tree. This
repeated structure is referred to as **fractals.
**

A **Sierpinsk triangle **is a simple example.
In the centre of an equilateral triangle, another equilateral triangle is drawn
such that the larger triangle is divided into four smaller triangles. The
central triangle is then cut out and the same is done to the remaining three
triangles as was done to the first one, this procedure repeating itself. This
gives us a final drawing whose circumference is unlimited in length and which
has an unlimited number of vertexes. The resulting curve though cannot be
calculated using infinitesimal calculation.

When the French mathematician Gaston Julia studied the rational function

x = x2
+* i*

where x is a complex number and i an imaginary one

he obtained a function which, after attaining a certain value, produced an unpredictably chaotic value. In this way, he obtained self-similar values which were called, after him, Julia’s values.

Based on Julia’s values, Benoit Mandelbrot created a computer program which was supposed to graphically describe this and which made him famous. The values which were a basis for this were called Mandelbrot’s values.

We can repeatedly find the shape of fractals in many geographic and biological shapes. Clouds, mountains and shorelines often have similar, general characteristics.

Fractals play an important role in computer graphics and are used, for example, to compress video games.

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