[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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