(a) Input-entropy oscillations with time, wavelength=7 time-steps, Left edge: superimposed histogram values plots. (b) The entropy-density scatter plot --- input-entropy (y-axis) against the non-zero density (x-axis), for about 33000 time-steps. (c) The same plot for just a few pulsing cycles, but linking successive dots giving a time-history. (a) Value-density oscillations with time. 0=green, 1=brown, 2=black. (b) The density return-map scatter plot --- the density of each value at t (x-axis) against its density at t+1, plotted as colored dots, for about 33000 time-steps. (c) The same plot for just a few pulsing cycles, but linking successive dots giving a time-history. EDD#32.12.1 While the "entropy-density plot" or the "density return map" is active, on-the-fly key-hit , (the coma) will toggle the connection of successive dots of the scatter plot for a linked history, as in the examples above for the v3k7 Spiral‑rule. The linked plot gives a much faster picture of the characteristic "strange attractor" waveform in just a few time-steps.

Examples of random wiring in a 40x40 2D lattice. A bigger lattice would normally be required for robust pulsing, to avoid reaching a uniform value attractor.

The cellular automaton pulsing model (preprint) described the surprising phenomenon of spontaneous, sustained and robust rhythmic oscillations, pulsing dynamics, when random wiring is applied to a 2D ``glider'' rule running in a 3-value totalistic CA. The characteristic wave-forms are robust and depend on the specific glider rule applied. EDD#32.7.3 While running space-time patterns, a new on-the-fly key-hit ? (question mark) allows nonlocal (random) wiring to be re-randomised at each time-step. The network will be re-randomised according to any random wiring biases originally set in EDD#11 and EDD#12. It turns out that re-randomising makes no difference to the characteristic pulsing waveform. New complex "glider" rule collections are included in the updated dd_extra.tar.gz file. Activate a "glider" rule with on-th-fly with keyhit g. The collections include these 2d neigborhoods for k=3 to 7.

Wave-length wl for rule v3k7 g35 in \cite{Wuensche-pulsingCA}, with steady but variable wl oscillations, and jagged stretches on the downslope of the entropy plot profile (a) can give false min/max results, but this is fixed by increasing the time-step window to w=20) to smooth the plot (b). (c) shows typical data, with wl between 62 and 113 time-steps, though the actual range is slightly greater. The last line shows average values.

EDD#32.12.6.2 If pulsing is present and the "entropy-density plot" is active, a new on-the-fly key-hit " (inverted coma) will toggle automatic data in the terminal to identify the pulsing wavelenght and waveheight. On the left, typical data output is shown where the wave length is automatically recognised, despite a jagged plot profile and irregular wavelength. To save the data, interrupt with q,then enter d for as described in EDD#32.12.6.2. Examples of the 2d space-time patterns of v=3 glider rules that pulse when the local CA wiring is randomised. These are from the "glider" rule collections, where gx is the rule index that can be selected in EED#32.16. Cell values: 0=white, 1=red, 2=black. Green trails of 5 time-steps indicate glider velocity.

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Last modified: Nov 2018