Pearson's Classification (Extended) of Gray-Scott System Parameter Values  

The first fourteen (R, B, and alpha through mu) were defined by Pearson [1]; the others were added by me [2].

Type R

Evolves to a uniform red state.

Type R patterns belong to Wolfram class 1.

Examples: (F=0.014, k=0.057), (F=0.074, k=0.069).

Type B

Evolves to a uniform blue state.

Type B patterns belong to Wolfram class 1.

Examples: (F=0.050, k=0.059), (F=0.078, k=0.059).

Type alpha (α)

Spatial-temporal chaos composed mainly of wavelets and "fledgling spirals" that repeatedly grow and/or multiply and quickly annihilate upon hitting another object.

Type α patterns belong to Wolfram class 3.

Example: (F=0.014, k=0.053).

Type beta (β)

Spatial-temporal chaos with localized red and blue state in different spots at different times. This looks like waves on a blue ocean with periodic red "voids" that open up suddenly and then quickly fill in with blue.

Type β patterns belong to Wolfram class 3.

Examples: (F=0.014, k=0.039), (F=0.026, k=0.051).

Type gamma (γ)

Stripes, either wormlike or branching, with endless instability in the form of occasional changes due to overcrowding, grain boundary instabilities, or other localized events.

Type γ patterns belong to Wolfram class 3.

Examples: (F=0.022, k=0.051), (F=0.026, k=0.057).

Type delta (δ)

Includes true Turing patterns and many parameter values that produce similar patterns through a presumably related effect.

The true Turing pattern arises from a starting pattern that is all blue, with arbitrarily small noise or other irregularities.

"Turing-similar" patterns cannot arise from an all-blue starting pattern, instead they need to be "seeded" by a large amplitude disturbance (any non-blue-state area in the starting pattern will usually do) and the pattern grows outward from the disturbance.

In either case, the resulting pattern is a hexagonal array of negative spots, possibly with some stripes in place of rows of spots, and with grain boundaries that are asymptotically stable.

Type δ patterns belong to Wolfram class 2-a.

Examples: (F=0.030, k=0.055), (F=0.042, k=0.059).

Type epsilon (ε)

Spatial-temporal chaos composed mainly of spots that continually crowd each other out; after regions are opened up by die-outs; other spots on the boundary of the newly opened region grow via mitosis to fill in the space.

Type ε patterns belong to Wolfram class 3.

Examples: (F=0.014, k=0.055), (F=0.022, k=0.059).

Type zeta (ζ)

Very similar to type epsilon (ε), but the disturbances that cause spots to die are limited to a small percentage of the domain at any given time, and these areas of disturbance travel slowly. Wolfram class 3.

Type eta (η)

Mixture of spots and worms (stripes with two free ends and no branching) with ongoing activity at worm tips, where spots occasionally split-off and rejoin the worms.

Type η patterns belong to Wolfram class 3.

Example: (F=0.026, k=0.057).

Type theta (θ)

Stripes in isolation grow width-wise; final state is mostly all stripes, usually with branching and can be fully connected (network of loops).

Type θ patterns belong to Wolfram class 2-a.

Examples: (F=0.030, k=0.057), (F=0.038, k=0.061).

Type iota (ι)

Negative spots (negatons) with molecule-like interaction; solitary negatons are not viable. This pattern class is closely related to type pi (π).

Type ι patterns belong to Wolfram class 2.

Example: (F=0.046, k=0.0594).

Type kappa (κ)

Stripes in isolation grow width-wise; final state is mostly all stripes, usually in multiple disjoint sets (hedgerow mazes).

Type κ patterns belong to Wolfram class 2-a.

Examples: (F=0.050, k=0.063), (F=0.058, k=0.063).

Type lambda (λ)

Solitons that grow by mitosis (cell-division). After the space is filled, solitons rearrange into hexagonal grids with grain boundaries.

Type λ patterns belong to Wolfram class 2-a.

Examples: (F=0.026, k=0.061), (F=0.034, k=0.065).

Type mu (μ)

Stripes that grow from each end (worms), possibly also co-existing with inert (non-mitotic) solitons. After the space is filled by the worms, they reorganize towards parallel stripes and remain disconnected from one another.

Type μ patterns belong to Wolfram class 2-a.

Examples: (F=0.046, k=0.065), (F=0.058, k=0.065).

Type nu (ν)

Inert (non-mitotic) solitons. The number of solitons depends on the number and size of blue areas in the starting pattern. All large blue areas shrink and/or split up into solitons. The solitons then drift apart from each other to spread as uniformly as possible across the space, but this takes a period of time proportional to eKw where w is the width of the domain in lu and K is a constant, K≈20+1000F. In the following images (with F=0.046, k=0.067) each successive image represents approximately twice as much elapsed time:

t=76 tu
t=76 tu
t=153 tu
t=153 tu
t=306 tu
t=306 tu
t=612 tu
t=612 tu
t=1224 tu
t=1224 tu
t=2448 tu
t=2448 tu
t=4896 tu
t=4896 tu
t=9792 tu
t=9792 tu
t=19584 tu
t=19584 tu
t=39062 tu
t=39062 tu
t=78125 tu
t=78125 tu
t=1.5625×105 tu
t=1.5625×105 tu
t=3.125×105 tu
t=3.125×105 tu
t=6.25×105 tu
t=6.25×105 tu
t=1.25×106 tu
t=1.25×106 tu
t=2.5×106 tu
t=2.5×106 tu
t=5×106 tu
t=5×106 tu
t=1×107 tu
t=1×107 tu
t=2×107 tu
t=2×107 tu
t=4×107 tu
t=4×107 tu
t=8×107 tu
t=8×107 tu
t=1.6×108 tu
t=1.6×108 tu
t=3.2×108 tu
t=3.2×108 tu
t=6.4×108 tu
t=6.4×108 tu

In the last frame the solitons have stopped only because double-precision arithmetic is no longer adequate to accurately compute u+∂u.

Type ν patterns belong to Wolfram class 2.

More examples: (F=0.054, k=0.067), (F=0.082, k=0.063).

Type xi (ξ)

Large, sustained spirals similar to the (Belousov-Zhabotinsky) reaction in a Petri dish. The seed of the spiral is an essential and sometimes rare feature, which must exist in suitable quantities to produce a long-lived pattern. This means that if the domain is small, the spirals will die out; in this case the pattern becomes uniform red state. If the domain is large enough, occasional irregularities usually geerate new spiral seeds and thereby keep up the population of spiral seeds.

Type ξ patterns belong to Wolfram class 3.

Examples: (F=0.010, k=0.041), (F=0.014, k=0.047).

Type pi (π)

Type pi supports stripes, loops, and spots, all of them negative. These form stable localized structures, both stationary and moving, and localized force-like interactions whose nature (attractive or repelling) oscillates in sign with increasing distance and whose strength decreases exponentially with distance. The combination of this force interaction and motion means that you also get rotating patterns.

Most significantly, the type pi patterns display such a great diversity that it is impossible to characterize the final outcome of a system just by looking at its initial state, except for a limited class of starting states (e.g. when the starting pattern consists only of spots).

One specific example occurs at F=0.06, k=0.0609. I have explored this region extensively and written a paper which can be found here. There is also a catalog of patterns, several additional animations, and some discussion here.

Type π patterns belong to Wolfram class 4 (and possibly 4-a).


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References

[1] J. Pearson, Complex patterns in a simple system, Science 261 (1993) 189-192.

[2] R. Munafo, Stable localized moving patterns in the 2-D Gray-Scott model (2010) draft here (PDF) and figures


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