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A phase-space is an N dimensional space whose every point fully characterizes
the state of a system. The phase space of the water-wheel
system can be characterized
by three variables^{3.2}. If we
plot every state of the system in time according to these three variables,
we come up with a trajectory which characterizes the behavior of the system
over time. Now, if we would be able to predict how this trajectory moves
in the phase space, we would be able to predict the behavior of the system.
If the trajectory was a line moving on a simple 2 dimensional surface
and the system was linear, by having three samples of this trajectory we
would be able to predict the behavior of the system.
However, there are places in the phase space where the trajectory seems to
trace a very thin volume, and the volume is created by infinite
stretching and folding of a surface. The shape underlying this
strange ``surface'' is the cantor set, which is self-similar.
This implies that the trajectory is moving on a shape with infinite amount of
detail, meaning that a different direction could be taken according
to infinitesimal differences in initial condition. In a linear system, a
small error in initial condition could only cause an error proportional
to the original error. However in a system like this, an error (i.e., our
inability to measure conditions with infinite precision) could cause
completely different directions to be predicted for the trajectory.
It is important to note that there is no noise introduced into the system,
and this interesting behavior can be seen on the computer by trying to
predict the trajectory of the system by using the three differential
equations characterizing the system. By changing the integration interval
and initial conditions, we obtain completely different results, while we
are usually used to obtaining more accurate results when we integrate
over smaller segments of time.

** Next:** Fractional Dimensions
** Up:** What is Self-similarity?
** Previous:** What is Chaos?
** Contents**
Shahrokh Yadegari
2001-03-01