The Cook Book

Recipe for the week of January 8 - 14

CAPOW! version 2: Rudy Cooks Again

This week we are happy to announce the availability of version 2 of Rudy Rucker's CAPOW! software for Windows-based 1d CA simulation. Rudy's program, developed with support from the Electric Power Research Institute (EPRI) of Palo Alto, is distinguished especially by its support for various continuous- valued cellular automata. These are discretizations of so-called coupled lattice maps, combining features of traditional CA rules with those of the iterated maps of mathematical chaos theory. You can download the new CAPOW! from our Kitchen Sink, or by clicking on capow2_0.zip. See also the companion file capow2_0.txt.

This week's soup is of a new variety that Rucker calls 'Adjustable Ulam,' based on a 1955 paper of E. Fermi, J. Pasta, and S. Ulam, and designed to emulate a nonlinear wave pde. I was definitely sous-chef on this one; Rudy's email instructions were to boil until 'after the mountain ranges appear, but before the sky-sushi is gone.' How did I do? Since only the Master Chef is really fit to judge, the rest of this week's recipe is taken from the Fermi-Pasta-Ulam CA Types entry of the CAPOW! Help file ...

This CA is similar to the Wave Equation CA, with the addition of a non-linear (quadratic) term. The size of the non-linear term is regulated by the Non-linearity parameter, which appears in the top of the Electrical Powers CA Group when you select a Fermi-Pasta-Ulam CA type.

Version 2.0 includes a variation of Fermi-Pasta-Ulam called Adjustable Ulam. The F-P-U CAs are very interesting, but it is easy for them to become unstable. When such a CA becomes unstable it will generate a pattern of red, blue, and black triangles similar to the pattern obtained by a Standard digital Type 3 CA. If you look at an unstable F-P-U in graph mode, you can see that the intensity values are ricocheting back and forth between the maximum (red) and the minimum (blue) values, with a few stops near the middle (black). It is interesting that the unstable nonlinear continuous-valued CA does indeed produce patterns just like a Standard CA.

To make an F-P-U CA behave stably, try cranking the Nonlinearity all the way down to 0 by repeatedly clicking the left button of the Nonlinearity control. It settles down right away into a wave. Now you can boost the nonlinearity by clicking the right Nonlinearity button a few times. If you do this long enough, instability starts up again, and growing triangles of instability will appear. If you work very fast you can quickly click the nonlinearity back down to 0 to restore the smoothness. If you are watching this in the Scroll Up viewmode, it is a bit like seeing rocks (the unstable triangles) forming turbulence in a flowing stream (the stable parts of the simulation). When a Fermi-Pasta-Ulam CA is behaving stably with a non-zero nonlinearity parameter, it is still a sensitive kind of thing. If, for instance, you resize the window by dragging the right edge to make the window wider, it is likely that instability will pour in from the right side. This is likely to happen when you load a saved F-P-U CA. To remedy it, you can try temporarily lowering the Nonlinearity, perhaps then reseeding the pattern with a wave, and then inching the Nonlinearity back up.

The non-linear term takes the form of a force proportional to the square of the difference between neighboring cell intensities; whereas the normal linear wave just has a force proportional to the difference between neighboring cell intensities. With proper tweaking, you can get the Fermi-Pasta-Ulam CA to produce "solitons," which are wave-like disturbances that move in isolation, instead of as part of a train of repeated undulations. The Adjustable Ulam wave essentially does this tweaking process automatically in each cell. If a cells values become maxxed out, the nonlinearity parameter for that cell is set to 0. In succeeding updates, the nonlinearity parameter for that cell is increased a slight amount until either it reaches the user-set target value of nonlinearity, or instability sets in for that cell again.

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