Stationary Turing Structures

Figure 2. A new route to reach to a stationary structure (the Turing structure) via self-duplication of dots. Gray-Scott model (10) is usedfor calculations. The last panel is still not symmetrical because of the influence of noise initially added randomly at each pixel...

Castets et al. [31] have defined Turing structure as follows These are stationary patterns originating with Sole coupling of reaction and diffusion processes. They... 

When a reaction is carried out under the condition where the thickness of the solution mixture is less, the colored spatial patterns arise due to the variations in chemical concentrations and this pattern depends on the lime. The other important pattern which is stationary in time and periodic in space, or periodic in both time and space is known as Turing structures. Turing pattern looks like very beautiful and has close resemblance with biological and chemical systems. 

Perraud, J, J. Agladze, K. Dulos, E, De Kepper, P. 1992, Stationary Turing Patterns versus Time-Dependent Structures in the Chlorite-Iodide Malonic Acid Reaction, Physica A 188, 1-16. 

Straigthforward analysis of the roots of Equation( 5), shows that in a two-variable system, a Turing instability is always associated with a real eigenvalue, so that at onset the arising structure is stationary. When more variables are involved, this is not necessarily the case and wave instabilities associated with a complex eigenvalue can also appear. These generalized Turing structures [17] are beyond the scope of this paper. 

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)... 

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. 

Some of spatio-temporal patterns and time-evolving patterns in dissipative structures can be frozen to obtain stationary structure. Addition of Ag ions in the BZ reaction records the spiral pattern (8), and quenching the photoinduced phase separation in a polymer mixture results in a Turing-type structure (9). 

Such oscillations are observed in the form of chemical waves or stationary structures (Turing type) in an oscillatory system (Chapter 10). The relative magnitudes of diffusion... 

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