Coupled Cellular Automata: Symbiotic Patterns and Synchronization

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Modeling of cooperative behavior has great relevance to a wide range of scientific fields. Often trivial systems with primitive standalone behavior produce rich dynamics working in collaboration. This Demonstration allows you to couple up to four different one-dimensional elementary cellular automata (1D-eCA) and produce behaviors beyond those of single 1D-eCA. Patterns of these 1D-eCA in the standalone case are shown in four rectangles on the left. They can be compared to the two patterns of coupled evolution on the right (the rectangles labeled -output and
-output). Often, simple evolution of standalone 1D-eCAs results in rich coupled dynamics of
and
. Such behavior is "symbiotic", since the coupled behavior of
and
cannot be obtained from standalone 1D-eCA. Another feature of coupled behavior is "synchronization". Namely, the pattern of
is generally different from the pattern of
, yet often similar "twin" sub-patterns can be noticed in
and
. In all images, the time of evolution goes from top to bottom. The four images on the left are just samples of standard 1D-eCA Wolfram rules. These four images are not used directly to create
and
patterns and are given only for comparison with
and
. Browse the snapshots for interesting patterns. See the "Details" section for a more technical discussion of the implemented coupling.
Contributed by: Vitaliy Kaurov (March 2011)
Open content licensed under CC BY-NC-SA
Snapshots
Details
Controls:
zoom—increase or decrease the size of the initial condition and the length of evolution
seed—randomize the initial condition
and
—choose functions (multiplication "×" or addition "+" modulo 2) that couple evolution of
and
outputs (
and
in the formula below)
Coupling formula description:
,
.
Each parameter ,
,
,
can be
or
.
and
are coupling functions; each can be set to element-by-element multiplication "×" or element-by-element addition "+" modulo 2 of the vectors
and
. The result is interpreted again as an IC and repeatedly fed back into the same formulas. The evolution of
and
is shown in vertical rectangles on the right, labeled "
-output" and "
-output".
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