Space-Time Patterns

We begin witli tiie simplest CA of all namely those whose site variables take on only one of two values and evolve according to rules which depend only on the previous values of a given site and those of its immediate left and right neighbors  

We recall, from equations 2.8 and 2,9, that only 32 so-called legal rules (out of the 256 total possible rules that can be defined), leave the zero-state a — 0 unchanged and are such that reflection-symmetric local states - 100 and 001 , for example - yield the same values. We now examine the patterns generated by these types of rules. 

Several distinct classes of patterns, or behaviors, are evident  

Patterns of this third class in fact demonstrate a complex form of scale-invariance by their self-similarity, in the infinite time limit, different magnifications observed at the same resolution are indistinguishable. The pattern generated by rule R90, for example, matches that of the successive lines in Pascal s triangle ai t) is given by the coefficient of in the expansion of (1 - - xY modulo-tv/o (see figure 3.2). 

While the patterns generated by rules R18 and R218 are essentially the same as 

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Figure 1.3 shows a few examples of the kinds of space-time patterns generated by binary (k 2) nearest-neighbor (r = 1) in one dimension and starting from random initial states. 

Fig. 3.1 Space-time patterns of a few elementary legal rules (A = 2, r = 1).

Figure 3.1 shows the evolution of a few legal rules, starting from an initial state consisting of a single nonzero value at the center site. In each case, and all such one-dimensional space-time patterns appearing in this book, the time axis runs from top to bottom and sites with value ctj — 1 are colored black. 

Difference patterns are space-time patterns of the difference between two evolutions of the same rule starting from two different initial states. For example, the... 

Fig. 3.16 Space-time pattern of fc — 2, r — 1 rule R18 kink-nites (i.e.. neighboring a = 1 site.s) are indicated in solid black. Notice the stochastic-like kink trajectories, despite the strictly deterministic rule.

Blocked Space-Time Patterns The intrinsic phase structure of a rule can sometimes be directly observed by graphing its blocked pattern. We can describe the space-time sets of the two ordered states of rule R18 by... 

Figures 3.10-3.15 present some qualitative evidence for the self-organization of space-time patterns emerging out of initial configurations of uncorrelated sites. In this Section we introduce some of the quantitative characterizations of selforganization in elementary r = 1, k = 2 rules by examining these systems from two different points of view.

Although p t) and C ir) do give some insight into the behavior of complex rules, even a casual glance at the earlier space-time patterns reveals a rich diversity of structure that these measures are incapable of capturing. 

For noiiadditive rulas, can no longer be obtained by merely looking at the evolution of the initial difference state. A fairly typical nonadditive behavior is that of rule R126. We see that, apart from small fluctuations, H t) tends to steadily increase in a ronghly linear fashion. This means that as time increases, the values of particular sites will depend on an over increasing set of initial sites he., space-time patterns arc scnsitivr.ly dependent on the initial conditions. We will pick up this theme in our diseus.sion of chaos in continuous systems in chapter 4. 

Similarly, if the initial state consists of nothing but infinite repetitions of some invariant block of values, the space-time pattern will again be periodic. Figure 3.28, for example, shows sections of two infinite periodic patterns for elementary class rule R30, starting from the states -OlOl- and -OOlOOllOOlOOll- ... 

P ig. 3.29 Space-time pattern of one-dimensional elementary rule R22. 

Consider, for example, the space-time pattern resulting from application of to a single nonzero site value, shown in figure 3.30 We make the following observations ... 

Figure 3.31 shows sample evolutions for p = 0, 1/4 and 3/4. The space-time pattern for p = 0 rapidly settles into an ordered state consisting of checkerboard-pattern domains, separated by two-site kinks once formed, the kinks remain locked in place. As p is slowly increased, these kinks begin to undergo annihilating random walks, much like the ones we saw earlier in the evolution of (the deterministic) rule R18. Their density decreases like pkink Grassberger, et.al, observed... 

In fact, following a small initial transient period, temporal sections of the space-time pattern are always of the form... 

Fig. 3.36 Space-time pattern of fc = 2, r = 3 tot llistic rule T88 i.e. Park s Glider Gun.

Figures 3.38 and 3.39 show typical space-time patterns generated by a few r = 1 reversible rules starting from both simple and disordered initial states. Although analogs of the four generic classes of behavior may be discerned, there are important dynamical differences. The most important difference being the absence of attractors, since there can never be a merging of trajectories in a reversible system for finite lattices this means that the state transition graph must consist exclusively of cyclic states. We make a few general observations.

Fig. 3.38 Space-time patterns for a few one-dimensional r = 1 reversible rules starting from simple initial states.

Recall that difference patterns are simply the space-time patterns of the difference between two evolutions of the same transition rule starting from two different starting configurations. For example, the value of the T site at time t of a difference pattern for a k = 2 global rule and two different initial global states cti(0) and... 

Figures 3.58 and 3.59 show some typical space-time patterns of two-dimensional rules starting from random initial states. The figures show snapshots of runs taken at times i = 1,5,10,25,50 and a cross-sectional view of the y-axis sites taken along an arbitrarily selected x site value for all times t = 1 through t = 50.

In particular, the last relation implies that if B,T —> oo with T/B fixed, then the information per site in a B x T space-time region, Iq = 5meas(B, T)/B —> 0. In other words, completely random space-time patterns for one-dimensional CA can never really be generated. 

Forget for a moment that you know that figure 12.11 shows the space-time pattern due to a well defined local deterministic rule, and that the underlying universe really consists of nothing but bits. Suppose you are told only that this figure represents some sort of alien physics, and that you may see as many different samples of this alien world s behavior as you wish. How are you to make any sense of what is really going on ... 

The space-time pattern of Cerenkov wavefront can be reconstructed during off-line analysis fitting relation 9. The reconstructed muon direction will be affected by indetermination on PMTs position (due to underwater position monitoring) and on hit time (PMT transit time spread, detector timing calibration,...). 

We have seen that delayed feedback can be an efficient method for manipulation of essential characteristics of chaotic or noise-induced spatiotem-poral dynamics in a spatially discrete front system and in a continuous reaction-diffusion system. By variation of the time delay one can stabilize particular unstable periodic orbits associated with space-time patterns, or deliberately change the timescale of oscillatory patterns, and thus adjust and stabilize the frequency of the electronic device. Moreover, with a proper choice of feedback parameters one can also effectively control the coherence of spatio-temporal dynamics, e. g. enhance or destroy it. Increase of coherence is possible up to a reasonably large intensity of noise. However, as the level of noise grows, the efficiency of the control upon the temporal coherence decreases. 

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