Cyclic states

Fig. 2.9 The state transition graph Gc, computed for a Tdim lattice consisting of iV = 4 points (with periodic boundary conditions), and totalistic rule T2 . The vertices labeled ti represent transient configurations those labeled cd represent cyclic states, and give rise to the formal cycle cum decomposition C[ j =[3, lj-f[2,2].

Figure 3.19, for example, shows a few of the cycles that emerge on a size N = 10 lattice for a system evolving according to the class c3 rule R122. In practice, and as suggested by this figure, many fewer iterations than the maximum = 2 are needed before a cyclic state is reached. 

Cyclic States which are configurations that appear in cycles and are repeatedly visited over the course of the evolution they are the attractor states of the system in the sense that they comprise the set toward which all evolutions ultimately lead. If all states are cyclic then the system is globally reversible. 

Figures 3,20, 3,21 and 3,22 show how the number of cyclic states (Nc), the average cycle length Crmave) and total fraction of states on cycles fc) changes as...

Fig, 3,20 Number of cyclic states cis a function of lattice size for a few representative one-dimensional elementary rules see text. 

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.

We can simplify this matrix by making the following two simple observations. First, since there cannot be predecessor states for garden-of-eden states, subma.-trices d>i, d>2 and d>3 must all be equal to zero second, since all cyclic states are predecessors only of other cyclic states, submatrices d>3 = d>6 = 0. thus really looks like... 

Since if is therefore an isomorphism, we have proven that any tree rooted on a cyclic state is identical to the tree rooted on the null configuration. I... 

The previous discussion has shown us how to calculate the total number of possible cyclic states. We also know, from Lemma 2, that all cycle lengths must divide the maximal cycle length Hiv obtain the exact number of distinct cycles and their lengths takes a little bit more work. If flw prime, we know that the only possible cyclic lengths are 1 and It can then be shown that only the null configuration is a fixed point unless N is some multiple of 3, it which case there are exactly four distinct cycles of length one. If Hat i ot prime, there can exist as many cycles as there are divisors of Although there is no currently known closed form... 

Although we will not worry about the precise tree structure, we note that for all graphs, and therefore all additive rules, every cyclic state is a root of a tree which is isomorphic to the null tree (the one terminating on S = 0) ... 

This follows from the fact that all trees are isomorphic each of the cyclic states has an... 

If the number of sites N < oo, we know that the system will eventually reach some cyclic state. Suppose we obtain a cycle of period T ct(0) -a ct(1) -a ff T—l) -A cf(0). Then, since we have just concluded that L must be strictly decreasing for cyclic states. 

Since we will be dealing with finite graphs, we can analyze the behavior of random Boolean nets in the familiar fashion of looking at their attractor (or cycle) state structure. Specifically, we choose to look at (1) the number of attractor state cycles, (2) the average cyclic state length, (3) the sizes of the basins of attraction, (4) the stability of attractors with respect to minimal perturbations, and (4) the changes in the attractor states and basins of attraction induced by mutations in the lattice structure and/or the set of Boolean rules. 

In these equations, = 0 is the bottom of the catalyst bed and Xx is the conversion in the flow direction from bottom to top, while X2 is the conversion in the opposite flow direction. Bunimovich et al. (1990) suggest using Eqs. (52) to (54) for an initial estimate of the temperature profiles in order to speed up conversion on integration of the full model equations in Table X. This step would only be taken if it were the stationary cyclic state profiles that are wanted. 

At cyclic-state the sequencing of batches might be different from the first sequence. This is usual in processes with a number of steps taking place in different vessels. 

Fig. 12.15 Water reuse network (cyclic-state) (Majozi et al., 2006)... 

Fig. 12.18 Resultant water reuse network with process vessels used for storage (cyclic-state)...

As mentioned early on in this chapter, the cyclic-state sequence is generally different from the first sequence. This means that the entire targeting procedure to get the... 

Fig. 3.10. A simplified cyclic state of non-metal elements related to biological systems.

There is not so much competition between organisms following these developments as there is specialisation since new secondary energy and chemical sources are best employed in different compartments, here largely different isolated cells, chemotypes. (Use was made of debris by cells from other organisms.) Separation of anaerobes, plant- and animallike aerobes, including different chemotypes, where their coexistence and cooperativity is more notable than competition, was an essential evolutionary step towards a cyclic state of the whole ecosystem. 

---