Chemical oscillators mixed-mode oscillations

In addition to tire period-doubling route to chaos tliere are otlier routes tliat are chemically important mixed-mode oscillations (MMOs), intennittency and quasi-periodicity. Their signature is easily recognized in chemical experiments, so tliat tliey were seen early in the history of chemical chaos. 

The next problem to consider is how chaotic attractors evolve from tire steady state or oscillatory behaviour of chemical systems. There are, effectively, an infinite number of routes to chaos [25]. However, only some of tliese have been examined carefully. In tire simplest models tliey depend on a single control or bifurcation parameter. In more complicated models or in experimental systems, variations along a suitable curve in the control parameter space allow at least a partial observation of tliese well known routes. For chemical systems we describe period doubling, mixed-mode oscillations, intennittency, and tire quasi-periodic route to chaos. 

In order to model the oscillatory waveform and to predict the p-T locus for the (Hopf) bifurcation from oscillatory ignition to steady flame accurately, it is in fact necessary to include more reaction steps. Johnson et al. [45] examined the 35 reaction Baldwin-Walker scheme and obtained a number of reduced mechanisms from this in order to identify a minimal model capable of semi-quantitative p-T limit prediction and also of producing the complex, mixed-mode waveforms observed experimentally. The minimal scheme depends on the rate coefficient data used, with an updated set beyond that used by Chinnick et al. allowing reduction to a 10-step scheme. It is of particular interest, however, that not even the 35 reaction mechanism can predict complex oscillations unless the non-isothermal character of the reaction is included explicitly. (In computer integrations it is easy to examine the isothermal system by setting the reaction enthalpies equal to zero this allows us, in effect, to examine the behaviour supported by the chemical feedback processes in this system in isolation... 

We also now know that complex oscillations evolve as simple limit cycles become unstable, bifurcating to more complex limit cycles. Only a small number of bifurcation sequences account for all known scenarios. We have seen examples of mixed-mode sequences (H2 -I- O2) and period-doubling cascades (CO -I- O2). A third route involving quasi-periodic responses is known and arises in some chemical system [88], but has not yet been observed in combustion systems (except in some special studies in which the ambient temperature or some other parameter is forced to vary in some sinusoidal or other periodic manner [89]). The important lesson then... 

Petrov, V., S.K. Scott K. Showalter. 1992. Mixed-mode oscillations in chemical systems. J. Chem. Phys. 97 6191-8. 

This example shows that mixed-mode oscillations, while arising from a torus attractor that bifurcates to a fractal torus, give rise to chaos via the familiar period-doubling cascade in which the period becomes infinite and the chaotic orbit consists of an infinite number of unstable periodic orbits. Mixedmode oscillations have been found experimentally in the Belousov-Zhabotin-skii (BZ) reaction 2.84 and other chemical oscillators and in electrochemical systems, as well. Studies of a three-variable autocatalator model have also provided insights into the relationship between period-doubling and mixedmode sequences. Whereas experiments on the peroxidase-oxidase reaction have not been carried out to determine whether the route to chaos exemplified by the DOP model occurs experimentally, the DOP simulations exhibit a route to chaos that is probably widespread in the realm of nonlinear systems and is, therefore, quite possible in the peroxidase reaction, as well. 

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). 

---