Experiments period-doubling

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. 

Since the first report of oscillation in 1965 (159), a variety of other nonlinear kinetic phenomena have been observed in this reaction, such as bi-stability, bi-rhythmicity, complex oscillations, quasi-periodicity, stochastic resonance, period-adding and period-doubling to chaos. Recently, the details and sub-systems of the PO reaction were surveyed and a critical assessment of earlier experiments was given by Scheeline and co-workers (160). This reaction is beyond the scope of this chapter and therefore, the mechanistic details will not be discussed here. Nevertheless, it is worthwhile to mention that many studies were designed to explore non-linear autoxidation phenomena in less complicated systems with an ultimate goal of understanding the PO reaction better. 

In order to gain some experience with the new concepts introduced above, we will now discuss the stability properties of the fixed points and periodic orbits of the logistic mapping. The following is also a more in-depth presentation of the period doubling scenario briefly discussed in Section 1.2. 

It is important to understand that these measurements are difficult. Since 5 = 5, each successive bifurcation requires about a fivefold improvement in the experimenter s ability to measure the external control parameter. Also, experimental noise tends to blur the structure of high-period orbits, so it is hard to tell precisely when a bifurcation has occurred. In practice, one cannot measure more than about five period-doublings. Given these difficulties, the agreement between theory and experiment is impressive. 

Period-doubling in the Lorenz equations) Solve the Lorenz equations numerically for cr = 10, b-j, and r = 148.5. You should find a stable limit cycle. Then repeat the experiment for r = 147.5 to see a period-doubled version of this cycle. (Whenplotting your results, discard the initial transient, and use the xy projections of the attractors.)... 

Figure 5.72. The INADEQUATE sequence and the corresponding eoherence transfer pathway. The experiment selects double-quantum coherence during the evolution period with a suitable phase cycle (analogous to the selection in the DQF-COSY experiment. Section 5.6.2). In doing so it rejects all contributions from all uncoupled spins.

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 variety of dynamical behavior observed in this apparently simple system is truly remarkable. As r is increased, there are Hopf bifurcations to simple periodic behavior, then period-doubling sequences to chaos, as well as several types of multistability. One of the most interesting findings, shown in Figure 10.7, is the first experimental observation (in a chemical system) of crisis (Grebogi et al., 1982), the sudden expansion of the size of a chaotic attractor as a parameter (in this case, r) is varied. Simulations with the delayed feedback of eq. (10.48) taken into account give qualitative agreement with the bifurcation sequences observed in the experiments. 

Each chaotic state C, consists primarily of what at first glance appears to be a stochastic mixture of the adjacent periodic states and thus, for example, consists of a mixture of Po and P, as Fig. 6 of ROUX et al. [23] illustrates. However, maps constructed from the time series clearly yield smooth curves, not a scatter of points. These maps indicate that behavior is deterministic, not stochastic. Moreover, it is difficult to imagine stochastic processes that would lead to period doubling, the universal sequence, and tangent bifurcations, yet all of these phenomena associated with chaos are found in one-dimensional maps and in experiments on nonequilibrium chemical reactions, as will now be described. 

Fig. 7. Period doubling sequence time series with periods T dlS s), 2Tq, 2 Tq, obtained in experiments on the BZ reaction. The quantity measured was the bromide ion potential. The dots above the time series are separated by one period. (From [24].)...

We have found that period doubling is fairly common in the BZ reaction. Many periodic states lose their stability through period doubling, but often the interval for even the period 2 state is less than 2%. Hence the doubling is easy to miss unless one uses a finer mesh than is usually in experiments on nonequilibrium chemical reactions. For example, if a period 2 state were observed over a 2% range in residence time, then the entire infinite period doubling sequence would occur in a residence time range of only about... 

Experiments show that the major periodic regimes PpP2,P3 etc. all lose stability to chaos by the mechanism of period doubling and that this happens in both directions, that is, towards either longer or shorter residence times. 

SELINQUATE (Berger, 1988) is the selective ID counterpart of the 2D INADEQUATE experiment (Bax et al., 1980). The pulse sequence is shown in Fig. 7.4. Double-quantum coherences (DQC) are first excited in the usual manner, and then a selective pulse is applied to only one nucleus. This converts the DQC related to this nucleus into antiphase magnetization, which is refocused during the detection period. The experiment has not been used widely because of its low sensitivity, but it can be employed to solve a specific problem from the connectivity information. 

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