Cross-shaped diagram model

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 ... 

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. 

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) ... 

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,... 

The left ventricle can be diagrammed approximately as shown in Figure 7. Because of this a number of investigators have modeled the left ventricle as a vibrating shell in the shape of a truncated ellipsoid of revolution with a circular cross section. Both thin and thick walled versions of the ellipsoid model have... 

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