Cross shaped phase diagram

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. 

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) ... 

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,... 

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"... 

Figure 5- Bistability and oscillation in a typical chemical system (see text), a) Simple bistability b) Effect (- -) of adding a feedback species c) Behavior of the system in b) as function of time d) Cross-shaped phase diagram.

---