Mixed mode oscillations

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 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. 

Fig. 7. Mixed mode oscillations in the Belousov-Zhabotinski reaction when it is farther from equilibrium than it is in Fig. 6.

Fig. 16 are only rarely strictly periodic, because usually rather small fluctuations in the external parameters are sufficient to trigger abrupt changes. However, in principle, mixed-mode oscillations belong to the category of multiple-periodic limit cycles. If the behavior is governed by two incommensurate frequencies, i.e., the ratio of two periodicities is an irrational number. This situation is denoted by quasiperiodicity and has been realized experimentally with periodically forced oscillations, as will be described next. 

Figure 16. Regions of different dynamic behaviors during the galvanostatic oxidation of H 2 in the current density-halide concentration plane. H, small harmonic oscillations MMO, mixed-mode oscillations (see Section 11.4) R, large-amplitude relaxation oscillations. (After Wolf etalJ )...

Figure 36. Experimental bifurcation diagram of the existence regions of different oscillation forms during galvanostatic H2 oxidation on Pt as a function of the Cu concentration. pN, period-N oscillation. (A) one stable steady state (B) small-amplitude oscillations (C) mixed-mode oscillations (D) large-amplitude period-1 oscillations. (After Krischer et al.

The second example is again taken from formic acid oxidation. In Ref. 88 two sequences of mixed-mode oscillations are described which were found when formic acid was oxidized at an elevated temperature (50 °C) at a rotating platinum electrode. The interesting aspect here is that the large amplitudes in the first sequence are as large as the small amplitudes in the second sequence. Hence, the period-1 state that separates the two sequences corresponds to the 1 state of the first sequence and the 0 state of the second one. 

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

A studyof a fairly simple model of an enzyme reaaion that exhibits chaotic behavior, the peroxidase-oxidase reaction, provides a good illustration of the role of circle map dynamics and mixed-mode oscillations in the transition to chaos. In the peroxidase-oxidase reaction, the peroxidase enzyme from horseradish (which, as its name implies, normally utilizes hydrogen peroxide as the electron acceptor) catalyzes an aerobic oxidation... 

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. 

Figure 8.1 Mixed-mode oscillations in the BZ reaction in a CSTR. (Adapted from Hudson et al 1979.)...

Mixed-Mode Oscillations and the Slow-Manifold Picture... 

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). 

Figure 8.5 Mixed-mode oscillations in model A with a = 0.001, yS = 0.03, = 0.12,...

Figure 8.8 Mixed-mode oscillations in the manganese-catalyzed BZ reaction.

Other mixed-mode oscillations of this type have been seen in the Bray reaction (Chopin-Dumas, 1978) and several chlorite-based oscillators (Orban and Epstein, 1982 Alamgir and Epstein, 1985a, 1985b). In Figure 8.9, we present a phase diagram that shows the progression of 1" oscillations in the chlorite-thiosulfate system. 

At this point, we will make a brief mathematical digression in order to look at a fascinating aspect of the mixed-mode oscillations found in the BZ reaction. 

Obviously, one cannot expect to observe an infinite number of generations on the Farey tree, but Maselko and Swiimey did find that when they were able to adjust their residence time with sufficient precision, they saw the intermediate states predicted by the Farey arithmetic, though after a few cycles the system would drift off to another, higher level state on the tree, presumably because their pump could not maintain the precise flow rate corresponding to the intermediate state. An even more complex and remarkable Farey arithmetic can be formulated for states consisting of sequences of three basic patterns (Maselko and Swinney, 1987). The fact that the mixed-mode oscillations in the BZ system form a Farey sequence places significant constraints on any molecular mechanism or dynamical model formulated to explain this behavior. 

Rinzel (1981) has studied a number of models for bursting behavior and has developed both an intuitive understanding of the origin of this phenomenon and a classification scheme that describes the different ways in which bursting can arise. The essence of his analysis, which is similar in concept to the development of Barkley s model for mixed-mode oscillations that we discussed in section 8.1, rests on identification of a set of slow and a set of fast processes. In neurons, the fast processes are associated with the generation of the action potentials, while the slow processes typically determine the period of the overall oscillation. The membrane potential provides the key link between the two sets of processes. 

Barkley, D. 1988. Slow Manifolds and Mixed-Mode Oscillations in the Belousov-Zhabotinskii Reaction, J. Chem. Phys. 89, 5547-5559. 

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