Chemical oscillators slow manifold

The oscillations shown in Figure 8.1 are of the mixed-mode type, in which each period contains a mixture of large-amplitude and small-amplitude peaks. Mixedmode oscillations are perhaps the most commonly occurring form of complex oscillations in chemical systems. In order to develop some intuitive feel for how such behavior might arise, we employ a picture based on slow manifolds and utilized by a variety of authors (Boissonade, 1976 Rossler, 1976 Rinzel, 1987 Barkley, 1988) to analyze mixed-mode oscillations and other forms of complex dynamical behavior. The analysis rests on the schematic diagram shown in Figure 8.2. 

Figure 8.2 Schematic representation of a slow manifold in the concentration space of a chemical reaction that exhibits mixed-mode oscillations. The trajectory shown has one large and three small extrema in X and Y for each cycle of oscillation. (Reprinted with permission from Barkley, D. 1988. Slow Manifolds and Mixed-Mode Oscillations in the Belousov-Zhabotinskii Reaction, . /. Chem. Phys. 89, 5547-5559. 1988 American Institute of...

We have spent a great deal of time discussing model A because we believe that the scenario that it presents—a fast oscillatory subsystem that is driven in and out of oscillation by a slower, coupled subsystem that moves between two states—is both intuitively comprehensible and chemically relevant. Moreover, it can be used to derive insight into other sorts of complex dynamical behavior, such as quasiperiodicity or chaos, as well. The slow-manifold picture is, of course, not the only way in which mixed-mode oscillation can arise. Another route to this form of behavior is discussed by Petrov et al. (1992). 

To the author s knowledge, with the exception of the benzaldehyde autoxidation oscillating reaction and the methylene blue catalyzed oxidation of sulfide by oxygen, all the new chemical oscillators discovered since 1980 are the result of the bistability-oscillation approach. This shows that, if bistability (or pleated slow manifold) is not of basic necessity for oscillatory behaviour, our method which realies on some particular type of relationship between bistability and relaxation oscillations, is presently the most efficient method to produce new chemical oscillators. 

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