Turing stationary

Transient Turing-like patterns in PA-MBO reaction (Polyacrylamide-methylene blue-sulphide-oxygen) system are obtained. Non-Turing stationary patterns in FIS reaction (Hexacyanoferrate (Il)-iodate-sulphite system) are obtained using an experimental technique similar to that used in CIMA reaction. These patterns develop through propagation of chemical fronts from the initial perturbation [54],... 

Waves of chemical reaction may travel through a reaction medium, but the ideas of important stationary spatial patterns are due to Turing (1952). They were at first invoked to explain the slowly developing stripes that can be exhibited by reactions like the Belousov-Zhabotinskii reaction. This (rather mathematical) chapter sets out an analysis of the physically simplest circumstances but for a system (P - A - B + heat) with thermal feedback in which the internal transport of heat and matter are wholly controlled by molecular collision processes of thermal conductivity and diffusion. After a careful study the reader should be able to ... 

The original condition for the development of a Turing pattern, namely that the inhibitor diffuses faster than the activator, can be generalized allowing also for transport processes different from diffusion, in which case this necessary condition becomes A stationary pattern with a characteristic wavelength may develop if the... 

Fig. 68. Typical dispersion relations, displaying the growth rate of perturbations, A(n), vs. their wave number, n, of an S-NDR system for three different homogeneous steady states [33]. The lowest curve depicts the case of a stable homogeneous state, the middle one is close to a Turing-type bifurcation in which a stationary structure with the integer wave number closest to the maximum of the curve is born. The up-most curve shows a situation for which the homogeneous state is unstable with respect to perturbations lying within the wavelength range n, for which X(n) > 0.

S-NDR systems (4DL inhibitor Turing-like structures (n > 1) standing waves, anti-phase oscillations with n = 1 or mixed-mode structures with n > 1 pulses stationary domains (n = 1 or n > 1)... 

The peak shapes of metal chelating analytes are often poor because metal impurities in the stationary phase behave as active sites characterized by slowo desorption kinetics and higher interaction energies compared to reversed phase ligand sites. This phaiomaion is typical of silica-based stationary phases [31] ultrapure silicas were made commercially available to reduce it. However, styrene-divinylbenzene-based chromatogripliic packings suffer from the same problem and it was hypothesized that metals may be present in the matrix at trace conditions because they were used as additives in the polymerization process they may have been c tured via Lewis acid-base interactions between the aromatic ring n electrons and impurities in the mobile phase [32]. 

Density functional theory has also been applied to the Cope rearrangement. Nonlocal methods, such as BLYP and B3LYP, find a single transition state with approximately 2 A. The barrier height is in excellent agreement with experiment. These first DFT results were extremely encouraging because DFT computations are considerably less resonrce-intensive than MRPT. Moreover, analytical first and second derivatives are available for DFT, allowing for efficient optimization of stmc-tures (particularly transition states) and the computation of vibrational frequencies needed to characterize the nature of the stationary points. Analytical derivatives are not available for MRPT calculations, which means that there is a more difficult optimization procedure and the inability to fully characterize structures. 

A catastrophe corresponding to the sensitive state (5.128) is sometimes called the Turing bifurcation. Apparently, the question what is the time evolution of the solution of (5.121) type on losing stability by the spatially homogeneous stationary state (x, y) may be settled only by examining the exact equation (5.116). The methods of investigation of this problem will be discussed in the next section. 

Note, however, that in the case of a lack of diffusion the stationary state was unstable, see equation (6.138) for D = 0. In this case the diffusion stabilized, for D < Dcr l, this state. Of considerably more interest is the case of a loss of stability on accounting for diffusion by the stationary state stable in the absence of diffusion. Such an effect of diffusion can be readily shown to be possible only in the case of a system of two equations with diffusion. A catastrophe of this type, called the Turing bifurcation, will be considered for the Brusselator. 

The state (a, b/a) loses stability on crossing by the system, with a continuous variation in the control parameters a, b, Dx, Dy, the sensitive state defined by the equalities in inequalities (6.165). A catastrophe involving the loss of stability of the stationary state (equations without diffusion) due to the effect of diffusion is called the Turing bifurcation. 

Alan Turing s paper entitled The Chemical Basis of Morphogenesis [440] ranks without doubt among the most important papers of the last century. In that seminal work Turing laid the foundation for the theory of chemical pattern formation. Turing showed that diffusion can have nontrivial effects in nonequilibrium systems. The interplay of diffusion with nonlinear kinetics can destabilize the uniform steady state of reaction-diffusion systems and generate stable, stationary concentration patterns. To quote from the abstract,... 

The Turing condition C4 = 0 for hyperbolic reaction-diffusion equations leads to exactly the same conditions as for the standard reaction-diffusion equation, namely (10.42) and (10.40). In other words, the Turing condition is independent of and Ty. If inertia in the transport is modeled by HRDEs, then the inertia has no effect whatsoever on the Turing instability to stationary patterns. 

The first unambiguous observation of a Turing instability in any experimental system did not occur until 1990. That year, the Bordeaux group found convincing evidence for Turing patterns in an in vitro system, the CIMA reaction (see Sect. 1.4.9). The gap of almost 40 yr between Turing s theoretical prediction of diffusion-induced instabilities and the experimental realization of stationary chemical pattern was caused by two main factors. 

The coefficients C K) of the characteristic polynomial det[J( )—X tl4] = 0 and the Hurwitz determinants A K) are easily obtained using computational algebra software such as Mathematic A (Wolfram Research, Inc., Champaign, IL) or Maple (Waterloo Maple Inc., Waterloo, Ontario). The th mode undergoes a stationary bifurcation when condition (12.41d) is violated, namely c K) = 0, as discussed in Sect. 1.2.3, see (1.36). In other words, a Turing bifurcation of the uniform steady state corresponds to c iki) = 0 with k 0, while the stability conditions (12.41) are satisfied for all other modes with k fej. The feth mode undergoes an oscillatory bifurcation when condition (12.41c) is violated, namely A K) = 0, as discussed in Sect. 1.2.3, see (1.38). A wave bifurcation of the uniform steady state corresponds to A k i) = 0 with k f/ 0, while the stability conditions (12.41) are satisfied for all otha modes with k few As discussed in Sect. 10.1.2, see (10.29), a wave bifurcation cannot occur in a two-variable reaction-diffusion system. 

Tbe second possibility, a stationary spatial or Turing instability, requires that det J(r) = 0,... 

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