CML

CML. cml Chemical Markup Language extension of XML with specialization in chemistry nmjw.xml-crrd.org 57-59... 

A SMILES code [22], MDL Molfile [50], or JME s own compact format (one-line representation of a molecule or reaction including the 2D coordinates) of created molecules may be generated. The created SMILES is independent of the way the molecule was drawn (unique SMILES see Section 2.3.3). Extensions to JME developed in cooperation with H. Rzepa and P. Murray-Rust also allow output of molecules in the CML format [60]. 

Figure 6.X7A A corrosion-product and deposit mound on a mild steel service water pipe honeycombed by small tubelike organisms. Each hole is approximately 0.01 in. (0.025 cml in diameter.

Number. Acetate. Alcohol. AlconoL in the Original CMl. Acetate. Alcohol. Alcohol in the Original CHI. Number. 

As mentioned above, CMLs are simple generalizations of generic CA systems. Confining ourselves for the time being to one-dimension for simplicity, we begin with a one-dimensional lattice of real-valued variables ai t) R whose temporal evolution is given by... 

Let us begin by considering the stability of homogeneous solutions and/or initial-conditions i.e. by considering the stability of a simple-diffusive CML when cri(O) = a for all sites i , where a is a fixed point of the local logistic map F(cr) = acr(l—cr). Following Waller and Kapral [kapral84], we first recast equations 8.23 and 8.24... 

Now let us consider the stability of the two systems around fixed points of /(cr), and therefore around homogeneous solutions of the CML. From chapter 4 we recall that (7 = 0 is a stable fixed point for a < 1 and cr = 1 - 1/a is a stable fixed point for 1 < a < 3. Let us see whether our diffusive coupling leads to any instability. 

We see from both equations 8.32 and 8.33 that the most unstable mode is the mode and that ai t) = 1 - 1/a is stable for 1 < a < 3 and ai t) = 0 is stable for 0 < a < 1. In other words, the diffusive coupling does not introduce any instability into the homogeneous system. The only instabilities present are those already present in the uncoupled local dynamics. A similar conclusion would be reached if we were to carry out the same analysis for period p solutions. The conclusion is that if the uncoupled sites are stable, so are the homogeneous states of the CML. Now what about inhomogeneous states ... 

A natural question is how does the local period-doubling behavior cf the logistic map translate to its CML-version incarnation Without loss of generality, let us consider the Laplacian-coupled version of the logistic-driven CML ... 

It turns out that, in the CML, the local temporal period-doubling yields spatial domain structures consisting of phase coherent sites. By domains, we mean physical regions of the lattice in which the sites are correlated both spatially and temporally. This correlation may consist either of an exact translation symmetry in which the values of all sites are equal or possibly some combined period-2 space and time symmetry. These coherent domains are separated by domain walls, or kinks, that are produced at sites whose initial amplitudes are close to unstable fixed points of = a, for some period-rr. Generally speaking, as the period of the local map... 

One of the most striking higher-level behaviors observed in CMLs is the diffusion of the kinks/anti-kinks that separate the different domains, a behavior that should remind the reader of our earlier discussion of the diffusion of local kinks induced by the deterministic elementary CA rule R18 (grass84a] (see section 3.1.2). Before looking at some examples, let us see how this comes about. 

Fig. 8.6 Examples of Kanoko s four regimes of CML behavior, listed in order of increasing a [kaiiekoSSaj.

Dynamical Entropy In order to capture the dynamics of a CML pattern, Kaneko has constructed what amounts to it mutual information between two successive patterns at a given time interval [kaneko93]. It is defined by first obtaining an estimate, through spatio-temporal samplings, of the probability transition matrix Td,d = transition horn domain of size D to a domain of size D. The dynamical entropy, Sd, is then given by... 

We recall from our earlier discussion of chaos in one-dimensional continuous systems (see section 4.1) that period-doubling is not the only mechanism by which chaos can be generated. Another frequently occurring route to chaos is intermittency. But while intermittency in low dimensional dynamical systems appears to be constrained to purely temporal behavior [pomeau80], CMLs exhibit a spatio-temporal intermittency in which laminar eddies are intermixed with turbulent regions in a complex pattern in space-time. 

Figures 8.8-a through 8.8-d show a few snapshots illustrating intermittent behavior. We have used the logistic-equation driven CML (equation 8.34) and set D = 0.25 and a = 3.83. Each figure shows 16 times overlapped after a certain number of iterations have elapsed (100 iterations in figure 8.7-a, 200 in figure 8,7-b, 300 in 8.7-c, and 500 in 8.8-d). Notice how, depending on the time of the snapshot, some regions are periodic and others are turbulent. Islands cf order tend to come and go as time progresses i.e. the local order is intermittent.

In their study of CMLs exhibiting a Pomeau-Manneville intermittency, Crutchfield and Kaneko [crutch87] have observed the following general behavior ... 

Let us consider the same Laplacian coupled CML as in the previous sections, but with / equal now equal to the one-dimensional circle map i.e. consider... 

At the risk of oversimplifying, there are essentially three different dynamical regimes of the one-dimensional circle map (we have not yet formed our CML) (I) j A < 1 - for which we find mode-locking within the so-called AmoW Tongues (see section 4.1.5) and the w is irrational (11) k = 1 - for which the non mode-locked w intervals form a self-similar Cantor set of measure zero (111) k > 1 - for which the map becomes noninvertible and the system is, in principle, ripened for chaotic behavior (the real behavior is a bit more complicated since, in this regime, chaotic and nonchaotic behavior is actually densely interwoven in A - w space). 

In this section, we follow Chate and Manneville s analysis of the following onedimensional CML system ... 

Fig. 8.10 First and secoiid order approxiinatioiis (/i and f 2) of Cliate and Maiineville s [chate88a] teiit-niap-like local CML function / (equation 8.45).

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