WHAT IS IT?

This is a model of a 2

D cellular automaton where each cell's state can take a real value
between 0 and 1 and the state

updating rule consists of a coupled chaotic map (in this
case the logistic map). Each cell 'reads' its own stat
e and the state of each of its four
neighbours and updates its state accordingly. The result is an ever

changing aperiodic
spatio

temporal pattern with large clusters of cells with states near each other; moreover,
the global pattern emerges from a random
initial condition, showing a more organized
structure.
HOW IT WORKS

Each cell starts by being in a random state between 0 and 1, and it updates its state at
each time step by applying the following rules:

"Read your current state (si(t))
;

Read your North, South, East and West nearest neighbours states (sj(t)) and take the
arithmetic average of their states (which we denote by sa(t));

Set your new state according to the rule given below."
si(t+1) = A [wi si(t) + (1

wi) sa(t)]
[1

(wi si(t) + (1

wi) sa(t))]
where, wi is a coefficient between 0 and 1, and represents the weight that a cell gives to
its own current state when determining its new state (and 1

wi is the weight each cell
gives to its neighbours). In the current
model we set wi to be equal to 0.5.
If we take A = 4 we get the well known logistic map's chaotic dynamics, for A < 4 we get
different types of dynamics, from fixed points to periodic dynamics.
HOW TO USE IT

There are two items on the i
nterface tab, one is a slider rule for the A parameter, you can
set it to different values than 4 and see what type of dynamic you get from there, the
other is a 'turn

on neighbours/shut

off neighbours button', it allows you to see the results
in the dynam
ics without any neighbour interaction and the dynamics with neighbour
interaction. If you start with the neighbours button in the off position and turn it on latter
you can see the effects of the coupling on the pattern. You can also see the graph of the
s
eries of the states of a patch previously chosen, to get the feeling of how the global
network of cells affects a given local cell.
THINGS TO NOTICE

Start the model with neighbour coupling on, and notice how an organized pattern
emerges
from a previously disorganized one.
Notice also the differences in the local dynamics and in the global pattern from the
uncoupled to the coupled case.
THINGS TO TRY

Do try:

Different values for the parameter A, to see the different
dynamics, and try to start with
very low values increasing the parameter progressively to higher values to see the
bifurcation structure;

Start the model without neighbour interaction, and watch it run for a while taking into
your attention not only the
absence of general structure but also the trajectory of patch
(1,5), then switch on the neighbours and see how a different pattern emerges and a
different type of trajectory for patch (1,5).
Notice also that each cell has a state defined by real numbers b
etween 0 and 1, and the
grey colours you see are the result of multiplying the cells' state by 10, if you multiply by
100, for instance, you get a multicoloured pattern. Try to change the code and multiply
the cell's state by 100 or even 1000, and try to i
dentify the patterns in these different
colour schemes (note that if you multiply the cells' state by 100 or 1000 you are actually
enhancing what might be called the "resolution" with which you observe the system, and,
thus you are really seeing more and m
ore details and more and more differences).
EXTENDING THE MODEL

You could try to change the weights that a given cell attributes to itself and its
neighbours, different neighbourhood structure, different non

linear maps.
One thing
that might be interesting to try would be to set the following colour scheme: if
the patch state is above a given state (for instance 0.5) set the colour blue, else, set the
colour green, you can proceed in this manner with more and more fine partitions of
the
unit interval to see how the different cells visit the different subintervals of a given
partition of the unit interval.
CREDITS AND REFERENCES

Some references on deterministic chaos and spatio

temporal complexity:
Blanc

Ta
lon, Jacques and Deniau, Laurent., PCA and CA: a statistical approach for
deterministic machines, Complexity International, Vol. 2, 1995
Blanc

Talon, Jacques, Effective Computation of 2D Coupled Map Lattices, Complexity
International, Vol.6, 1998
Deane,
Deane, J.H.B. and Jefferies, D.J., The Behaviour of Coupled Map Chains,
Complexity International, Vol.3, 1996
Cvitanovic, P., Universality in Chaos, 2nd Edition, Bristol: Institute of Physics
Publishing, 1993
Gottlieb, H.P.W., Properties of Some Generali
sed Logistic Maps with Fractional
Exponents, Complexity International, Vol. 2, 1995
Gulick, Denny, Encounters With Chaos, McGraw

Hill, 1992
Kaplan, Daniel, Glass, Leon, Understanding Nonlinear Dynamics, Springer

Verlag, New
York, 1995
Shibata, Tatsuo,
Chawanya, Tsuyoshi, Kaneko, Kunihiko, Noiseless Collective Motion
out of Noisy Chaos, obtained from website http://arxiv.org/pdf/chao

dyn/9812007
Stogratz, Steven H., Nonlinear Dynamics and Chaos, Perseus Books, 1994
Web site for Complexity Internation
al:
http://www.complexity.org.au/
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