# Understanding chaos and attractors

[From Peter Small (2004.05.12)]

For those who are completely lost on the concept of dynamical systems
and attractors, I've written this analogy that might provide a
glimmer of understanding:

A dynamical system is any system or object that changes over time.
The effect of these changes can be represented in a "state space"
where all the possible states for the system or object would be
represented in this space.

This can be imagined by considering an automobile being filmed from
an overhead helicopter while journeying around a town. If this
journeying is filmed for every possible journey the automobile could
make, the frames of this film would represent the state space of the
vehicle in terms of its location and orientation. In other words, the
state space, as recorded on film, would contain all possible
locations and orientations during the course of all possible journeys.

Any particular journey could then be described by referring to a
particular sequence of frames, or, a particular sequence of positions
and orientations. This would be know as a trajectory through the
state space, with a starting and a finishing position.

Now think of a die being thrown. It would bounce around a table and
finally come to rest on one of its sides. In this activity, the die
would move chaotically in an unpredictable way. A film made of all
possible throws would represent the state space of thrown dice -
showing all the possible positions and orientations for a die during
any throw.

Any particular throw would have a unique sequence of positions and
orientations as it bounced around before coming to rest. This would
be seen as a trajectory through the state space as it moved between a
unique sequence of its possible positions and orientations as defined
in the state space.

Clearly, every throw would have a unique sequence of positions and
orientations in the state space that described its dynamical
activity. Although this is a deterministic system (determined by laws
of physics), it wouldn't be practical to work out this sequence of
events to be able to predict which side of the die would be uppermost
when it came to rest.

For this reason, we never concern ourselves with what happens in the
state space, we only concern ourselves with the final resting
position, which produces one of six possible results: the die
randomly producing a number between one and six.

This illustrates the essence of chaos theory. An unpredictable,
dynamical system, acting chaotically, comes to a resting position in
one of six possible states when it runs out of momentum. In chaos
theory, each of these six possible states is described an an
attractor of the system. Thus the dynamical system of a thrown die
can be described as a chaotic system that has six attractors.

In a similar way, the unpredictable journey of a ball thrown onto the
spinning wheel of a roulette table will end up in one of thirty seven
possible attractors (thirty-eight if the wheel has a double zero
slot).

Now think of a casino with thirty-eight tables. Say you spun the
wheel on one table and then, depending upon which slot the ball fell
into, one of the other tables would be chosen for the next spin. Now
think of there being thirty eight of these casinos and this second
spin determined which of the casinos the next spin would be in.

If you now substitute neural networks for roulette wheels, you can
get some idea as to how attractors can be used to control where
different activity in the brain takes place. Each place of activity
brings a unique combination of other neural networks into play that
together produce a perception, a state of awareness and particular
emotions.

Of course, this is a very simplistic metaphor, but it illustrates the
essence of the way in which perceptions are constructed in the brain
as a consequence of attractors. In the brain the sequences are not
initiated by chance but by combinations of inputs from the senses and
the routes and connections to auxiliary networks are influenced and
modified by genetic structures and hebbian learning.

Peter Small

Author of: Lingo Sorcery, Magical A-Life Avatars, The Entrepreneurial
Web, The Ultimate Game of Strategy and Web Presence
http://www.stigmergicsystems.com

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