#32: Patterns A1, A2 and A3, shown in Figure 3, are generated by automata #32. Pattern A1 is generated by random order automaton, pattern A2 by an interlaced order automaton, and pattern A3 by the synchronous automaton. These automata generate patterns that die out almost immediately. However, the randomized automaton generates longer patterns. The synchronous version of automaton is classified to Class I (homogeneous automaton) by Wolfram [Wol 83].
(Similar examples are here.)
#100: There are no significant differences between patterns, B1, B2 and B3, shown in Figure 4, which are generated by automata #100. However, the randomized automaton generates the longest transient patterns, and the synchronous one generates the shortest. The synchronous version of automaton is classified to Class II (periodic automaton) by Wolfram [Wol 83].
(Similar examples are here.)
3.2 Fluctuated patterns
Patterns, which are fluctuated by randomization but whose major characteristics are preserved, are shown and their properties, such as their life-time, are argued in the present section. Patterns generated by automata #226, #146 and #22 (See [Ing 84]) are shown for example in Figures 5, 6 and 7.
#226: Patterns C1 and C2, which are shown in Figure 5, are generated by automata #226. The shapes of black or white domains in pattern C1 and those in pattern C2 are quite different. However, these patterns have the same characteristic. Many black domains (domains of 1's) grow first, then shrink and die in both patterns. However, the white-to-black borders, i.e., the borders whose left side is 0 and right side is 1, move like Brownian particles in the randomized case. The randomized automaton generates longer transient patterns. The final state of the interlaced case is determined solely by the initial state (by fate), but that of the randomized case is determined by both the initial state and the random numbers. The synchronous version, pattern C3, has less similarity to the others.[*3.1]
#146: Patterns D1 and D2 shown in Figure 6 are different in several points. Firstly, black domains are mortal in the random case, i.e., the final state is always uniformly white (0's). However, black domains or the slanting lines exist forever in the interlaced case, unless those move right and those move left exactly extinct in pair. Secondly, a border between a black domain and a white domain goes left or goes right at a constant speed in the interlaced case, but one can move left or right freely, depending on the order of computation, in the randomized case. These patterns are also much different from pattern D3, which is generated by the synchronous version and is chaotic.
#22: Both patterns, E1 and E2 shown in Figure 7, generated by automata #22 have stripes as their background. Particles or lattice defects moving left or right can be seen in both cases. These patterns are also much different from that of the synchronous version, pattern E3, which is chaotic. Particles can move in both directions, such as Brownian particles, in the randomized case. This pattern is very similar to a pattern generated by a (deterministic) CML (coupled map lattice) in ``diffusion of defect'' phase [Kan 89]. Particles can move either left or right in the interlaced case. If two particles crash, they always seem to disappear in the randomized case, but they often cross or reflect each other in the interlaced case. Particles can also crash and disappear in the latter case, but this type of interaction can occur only in the early stage.
3.3 Merging and/or splitting patterns
Patterns, in which domains are merging and/or splitting, generated by asynchronous automata are shown in the present section. Patterns generated by automata #166, #58 and #38 are shown for example in Figures 8, 9 and 10.
#166: The differences between the random and interlaced cases, F1 and F2 shown in Figure 8, are as follows. Firstly, two black domains sometimes merge into one in the former, but they do not in the latter. The final state is uniformly black (not white!) in the former. Secondly, black domains move left in both cases, but the speed of this motion is much slower in the former. However, they never move right. Pattern F3 has similarity to F2 in its stripes, but it is different from F2 in its cyclic, but nearly chaotic, behavior.[*3.2]
#58: The characteristics of both patterns, G1 and G2 shown in Figure 9, are completely different in this case. The differences are as follows. The two differences in automaton #146 are the same for automaton #58. Thirdly, a black domain sometimes splits into two, and two black domains sometimes merge into one in the randomized case. G1 is similar to some of those generated by synchronous automata of Class IV (complex automata) with more neighbors (a larger value of r) [Wol 84]. However, similar patterns are never generated in the interlaced case. G1 is similar to the pattern of automaton #50 [Ing 84]. However, the lifetime of G1 is longer. Pattern G3 has similarity to G2.[*3.3]
(Similar examples are here.)
#38: Patterns H1 and H2, shown in Figure 10, have similarity to those shown in Figure 8. However, pattern H1 is more complicated than pattern F1 because the black domains not only merge but also sometimes split into two domains. Pattern H3 has similarity to H2.
3.4 Chaotic or partially chaotic patterns
Chaotic and partially chaotic patterns generated by 1D-ACA are shown in the present section. Some 1D-ACA generate chaotic patterns, some of which is very similar to patterns generated by synchronous automata, and others are quite different from them. Patterns generated by automata #105, #57 and #60 are shown for example in Figures 11, 12 and 13.
#105: No significant structure can be seen in pattern I1 shown in Figure 11, which is generated by the randomized automaton. Thus, this is classified into chaotic patterns. However, in pattern I2 shown in Figure 11, which is generated by the interlaced automaton, waves moving to the left or right can be seen. Thus, this is orderly. The synchronous version, pattern I3, is chaotic, but quite different from I1.
#57: The patterns generated by automata #57, which are shown in Figure 12, are more complex, but they are similar to those of #1. The pattern is chaotic in the randomized case, J1. The pattern seems to be more orderly but still chaotic in the interlaced case, J2. The complex structure seen in the interlaced case and its noise-sensitivity is analyzed in Section 5.
(Similar examples are here.)
#60: Pattern K2, which is generated by interlaced automaton #60, is similar to pattern K3, which is generated by synchronous automata classified to Class III (chaotic automata) by Wolfram, and pattern K1 can be seen as a randomized version of K2. Interlaced automaton #60 is one of the rare asynchronous automata that generates Class-III-like patterns.
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Some automata generate partially chaotic patterns, such as those shown in Figures 14 and 15.
#73: The thick black stripes in patterns L1 and L2 shown in Figure 14 do not change nor move. The patterns between these stripes are chaotic in the randomized case, and white particles moving left or right can be seen in the interlaced case. Pattern L1 has a similarity to the pattern generated by automaton #108 [Ing 84]. Pattern L3 is also similar to L1 and L2, but the difference is that black stripes are dynamically generated in L3, but they are static in L1 and L2.
#37: Patterns M1 and M2, shown in Figure 15, are quite different each other. In the randomized case, fluctuated particles split and merge, and striped domains are stable. However, in the interlaced case, particles move straight and extinct in pair, and striped domains are unstable. The stability of dark domains in M2 seems to be a phantom. Pattern M1 is also similar to patterns generated by CMLs in ``diffusion of defect in chaotic media'' phase [Kan 89]. The difference between M1 and M2 is worth researching in detail in future. Pattern M3 is also quite different from M1 and M2.
