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Until recently signals were categorized as either being deterministic or
random. If a deterministic signal was an oscillating signal and had an
infinite amount of energy
it was supposed to be periodic. The discovery of chaotic systems
meant that this
assumption no longer holds. When Lorenz detected chaos,
he called it: ``Deterministic Nonperiodic Flow''[26].
Chaos was an observed phenomenon which went against the usual
scientific
intuition; obviously intuition is a highly subjective matter and one
should create ones own perception of this statement. Lorenz studied
the phenomenon of convection in fluids. However, his equations can
be mapped to a very simple mechanical system. Imagine a waterwheel with
many buckets connected to it (see Figure 33).
All the buckets have holes in the bottom so that the water can run out.
A steady flow of water is supplied from the top. If the wheel is started
with a small push, the buckets on the top are filled and by the time
they reach the bottom, they are mostly empty. Therefore, one side of
the wheel becomes heavier than the other. If we increase the flow of the
water the wheel starts to turn faster. Once we have passed a certain
threshold, the system can start to act chaotic. The wheel can turn so fast
that by the buckets which reach the bottom of the wheel are not
completely empty, and
the buckets that pass under the flow do not have enough time to fill up,
and the wheel starts to get slower, then it gets slow enough that
the original situation causes it to speed up again.
This oscillation becomes damped to the point that the
wheel starts to turn the other way around; this means that the oscillation of
getting faster and slower damps out at the point that if the wheel was
turning to the right, the left buckets would be heavier and the wheel starts
to turn in the other direction. What would happen if we let
such a system ``cool down'' without changing any parameters?
This is where the scientific intuition used to provide different answers
than nature. One may think that the system will eventually pick up a pattern,
however long this pattern may be, and keep repeating that pattern.
Lorenz showed that this system will never repeat itself, which means that
even though the behavior of the system is called deterministic (i.e., three
differential equations model the system), the resulting behavior is
nonperiodic. Lorenz explained such behavior by showing that the
phasespace of this system contains a space which is created from volumeless
surfaces with infinitely detailed structure.
Figure 33:
Waterwheel imitating the convection system of
Lorenz[14].

Next: Relationship Between Chaos and
Up: What is Selfsimilarity?
Previous: Introduction
Contents
Shahrokh Yadegari
20010301