U-Skate World, an Instance of the Gray-Scott System

The Gray-Scott system produces a wide variety of interesting patterns and behavior, such as cells that multiply, self-organizing fingerprint patterns and much more. Most of these have some variety, for example I have seen 4 different patterns of cell division.

However, one particular area of the parameter space exhibits patterns with at least as much diversity as Conway's Game of Life and similar systems. The images on this page show some of the more common patterns; there are also some illustrations for my 2009 paper and a fairly complete collection of stable patterns here: catalog of patterns.

I refer to this particular system (Gray-Scott using the Pearson constants and F=0.0620, k=0.0609) as U-skate world because of the skate, the U-shaped, common, persistent moving object seen in most of the examples below.

Negatons

The most common stable shape is a negaton, or negative soliton. A true negaton has a circular shape that can exist in any large blue (low U, high V) area. The more common "negative solitons" mentioned in most of my Gray-Scott parameter descriptions (such as here and here) are really just a roughly circular interior of a loop.

First true negatons

Negatons exist at many parameter values near the western (low k) edge of the complex region identified by Pearson. At the very westernmost parameter values they are stable (viable) only if they have one or more other negatons as neighbors. This video shows the behaviour at (F=0.0460, k=0.0594), described more fully here. It is the first place I found true stable negatons.

In the "U-skate world" parameters discussed below, negatons survive indefinitely in isolation, and also tend to clump together in groups. They exhibit a behavior similar to bubbles in water with "surface tension", moving more quickly when they get close to contact, and holding back the movement of other shapes such as those discussed below.

First convincing evidence of class-4 complexity

Above is a video of the behaviour at (F=0.0620, k=0.0609) that made me believe I would find phenomena as complex as a Wolfram "class 4" cellular automaton.

U-Skate Discovery Video

In the early morning of 2009 March 23^rd I saw the U-shaped moving pattern for the first time. This is the first simulation that I made a recording of. (2009 March 23^rd at 3:25 AM).

Skates

The skate or U-skate is a moving shape that keeps its form and moves indefinitely if unimpeded.

Mapping out the region

The above simulation shows a range of parameter values; higher F is near the top; higher k is near the right. Based on this and some other similar tests, I decided to move down to F=0.0600.

Early, Middle and Late-stage Evolution

Diverse behaviors over different time scales

This is an "accelerating time-lapse" video with a simulation rate that increases progressively: the simulated speed doubles every 7 seconds.

0:00-0:40 Fixed-rate growth : First we see normal growth of blue spots to mostly fill the space, leaving some spots and stripes and a lot of solid blue. Some stripes shrink to spots. This is very similar to the early evolution of Gray-Scott systems at many other F and k parameter values.

0:40-1:15 Complex behavior, first stage : Once the field is mostly blue with stripes and spots, the pattern starts to evolve in ways that exhibit both orderliness and unpredictability. Any remaining stripes gradually bend, alternately lengthen and shrink, and start to create parallel features. Pause the video at 1:12, and you can see a bunch of closely-spaced yellow stripes in the lower-left, and a pair of yellow stripes at a wider spacing in the bottom-center.

1:15-1:29 Complex behavior, second stage : At about 1:15 there are few solid blue regions left, and the system switches to a different type of behavior that is both orderly and unpredictable. Groups of parallel stripes "compete" with one another in complex ways. At times the space is completely filled by stripes and negatons; at other times areas of solid blue open up.

1:29-1:42 Stability : At 1:29, all chaotic activity immediately and permanently stops. This is very similar to the fate of large fields of chaotic activity in cellular automata like Conway's Game of Life. The final state contains stripes, negatons, and a region of solid blue, and it is qualitatively similar to the chaotic states that led up to stability.

The only remaining motion is a continuous "drift". This drift is very slow, but here appears to accelerate because of the accelerating time-lapse nature of the simulation. The direction of drift is related to the shape and orientation of the curved stripes.

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