Cross-shaped diagram

Another important reaction supporting nonlinear behaviour is the so-called FIS system, which involves a modification of the iodate-sulfite (Landolt) system by addition of ferrocyanide ion. The Landolt system alone supports bistability in a CSTR the addition of an extra feedback chaimel leads to an oscillatory system in a flow reactor. (This is a general and powerfiil technique, exploiting a feature known as the cross-shaped diagram , that has led to the design of the majority of known solution-phase oscillatory systems in flow... 

The very narrow region of the constraint space where oscillations could occur in this system was not found until after the chlorite-iodate-arsenite oscillator had been discovered. Its occurrence is shown in the cross-shaped diagram for chlorite-iodide (Fig. 8.)... 

Epstein and Kustin [39] set up a first reaction mechanism for the chlorite-iodide oscillator and reported a (diagonal) cross-shaped diagram in the species [CIO2 ]o-[I ]o- Later, Citri and Epstein [40] revised the first suggestion for the mechanism by reducing the original model considerably. Flere we employ the revised mechanism proposed by Citri and Epstein. The elementary steps are as follows ... 

DE KEPPER - The detailed analysis of cross shaped diagrams can be found in two papers ... 

In Fig. 6, separate regions of bi-stability, oscillations and single stable steady-states can be noticed. This cross-shaped phase diagram is common for many non-linear chemical systems containing autocatalytic steps, and this was used as an argument to suggest that the Cu(II) ion catalyzed autoxidation of the ascorbic acid is also autocatalytic. The... 

Figure4,7 The cross-shaped phase diagram for r 1/jit. Shown here are the four major regions and the critical values of A (solid lines) for the system of eqs. (4.2). Dashed curves are obtained from more detailed nonlinear stability analysis. (Adapted from Boissonade and De Kepper, 1980.)...

Thus, we have two autocatalytic reactions that have the species 1 and I2 in common. In late 1980, Patrick De Kepper, who had developed the cross-shaped phase diagram model while working with Jacques Boissonade at the Paul Pascal Research Center in Bordeaux, arrived at Brandeis University to join forces with Irving Epstein and Kenneth Kustin, who had independently come to the conclusion that chemical oscillators could be built from autocatalytic reactions and had targeted the chlorite iodide and arsenitc iodate systems as promising candidates. The collaboration quickly bore fruit. 

Figure4.11 Phase diagram of the chlorite odide reaction in a CSTR. Inset shows the same data on a linear scale, focusing on the oscillatory region. P indieates the cross point of this cross-shaped phase...

The examples of DDEs that we have considered so far have all been linear. Linear systems allow a considerable amount of analysis and even exact solution, on occasion, but few real systems are linear. We now turn to some examples that involve nonlinear DDEs, starting with two familiar examples, the cross-shaped phase diagram and the Oregonator. In these two models we see how, much like in the case of the sequence of first-order reactions treated above, one can reduce the number of variables by introducing a delay. In the nonlinear case, however, the choice of the delay time is far more difficult than in linear models. 

In Chapter 4, we discussed the role played by the cross-shaped phase diagram in the systematic design of chemical oscillators. A key element in that effort was the two-variable ODE model (Boissonade and De Kepper, 1980) ... 

Examples of the results obtained are given in the phase diagrams of Figure 10.5. Figure 10.5a illustrates the situation for the case of the usual cross-shaped... 

Figure 12.11 Schematic cross-shaped phase diagram (see text for discussion).

Epstein, I, R, Luo, Y, 1991. Differential Delay Equations in Chemical Kinetics. Nonlinear Models The Cross-Shaped Phase Diagram and the Oregonator, J. Chem. Phys. 95, 244-254,... 

A cross-sectional diagram of a planar triode is shown in Fig. 6.7. The envelope is made of ceramic, with metal members penetrating the ceramic to provide for connection points. The metal members are shaped either as disks or as disks with cylindrical projections. 

A consideration of CFT starts with two fundamental concepts the coulombic theory of electrostatic interactions and a detailed knowledge of the shapes of d orbitals. Two-dimensional cross-sectional diagrams of hydrogen-like orbitals show them as being cut out of circular pieces of cloth by various nodes. The sum of the electron probabilities in a given subshell is a sphere. The five d orbitals appear to be composed of four similar and one, the d, special orbital. To see that the dj. is not unique in shape or energy, a set of six dependent d orbitals is first visualized. The dj. orbital turns out to be just a linear combination of two of these dependent orbitals that look exactly like the other four. 

Sulfur chemistry has provided the other source of new non-halogen oscillators. BURGER and FIELD [36] found oscillations in the reaction of methylene blue, sulfide, sulfite and dissolved oxygen. The system, as illustrated in the phase diagram of Fig. 2, does not appear to exhibit bistability or a cross-shaped phase diagram, though it may instead possess a continuous range of steady states [37]. 

A second sulfur-based oscillator, the hydrogen peroxide-sulfide reaction recently discovered by ORBAN and EPSTEIN [38] shows, in contrast, an almost classic cross-shaped phase diagram. Fig. 3. The Burger-Field system is the first to oscillate in basic solution, while, as we see in Fig. 4, the H O -S oscillations go from acidic to basic pH, making possible dramatic color effects with acid-base indicators. 

Recent progress in discovering new chemical oscillators has been aided immeasurably by the development and use of systematic search procedures. By far the most productive such technique has the "cross-shaped phase diagram"... 

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