Other Oscillating Systems

As may well be apparent periodicity in chemical systems requires coupled, competing processes. Without coupling via bromide ion the Belousov-Zhabotinskii reaction would just provide parallel pathways for the consumption of bromate ion and malonic acid. The coupling compels feedback which creates the possibility of switching from one path to the other. Since the time scale of the processes is comparable switching and oscillation are observable. In a similar way thermochemical and electrochemical oscillators may be designed. Furthermore spatial, rather than temporal, periodicity may be observed. We consider a few examples. 

Zhabotinskii, Dokl. Akad. Nauk SSSR 157, 392 (1964) Biofizika 9, 306, (1964). 

Winfree, Science 175, 634 (1972) Sci. Amer. 230(6), 82 (1974). The latter article has particularly pretty photographs. 

A somewhat different interpretation postulates that precipitation is preceded by the formation of colloidal particles of the insoluble material. These then become charged by preferential adsorption of either anion or cation which, due to electrostatic interactions, is autocatalytic. The larger 

This curious phenomenon was discovered by G. Lippman, Ann. Phys. 2S 149, 546 (1873). 

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A thorough insight into the comparative photoelectrochemical-photocorrosion behavior of CdX crystals has been motivated by the study of an unusual phenomenon consisting of oscillation of photocurrent with a period of about 1 Hz, which was observed at an n-type CdTe semiconductor electrode in a cesium sulfide solution [83], The oscillating behavior lasted for about 2 h and could be explained by the existence of a Te layer of variable width. The dependence of the oscillation features on potential, temperature, and light intensity was reported. Most striking was the non-linear behavior of the system as a function of light intensity. A comparison of CdTe to other related systems (CdS, CdSe) and solution compositions was performed. 

Various irradiation parameters were investigated, such as the types of organic additives, intensity of the ultrasound, dissolved gas and distance between the reaction vessel and the oscillator. In the case of frequency effects, other irradiation systems were used. The details are described in the section of effects of ultrasound frequency on the rate of reduction. 

Figure 7 shows the dynamic responses obtained when the set point for the intermediate component was changed from 0.98 to 0.984. One may notice the better response provided by the Petlyuk column in this case, which is faster than the other two systems and without oscillations. When the lAE values were calculated, a remarkable difference in favor of the Petlyuk system was observed 2.87 x 10 for the Petlyuk column, compared to 0.0011 for the PUL system and 0.0017 for the PUV system. The results from this test may seem unexpected, since the new arrangements have been proposed to improve the operation capabilities of the Petlyuk column. The SISO control of the intermediate component, interestingly, seems to conflict with that of the other two components in terms of the preferred choice from dynamic considerations. 

Out-of-equilibrium systems (non-linear, dynamic processes), such as the Zabotinski-Belousov reaction, and other oscillating reactions bifurcation, and order out of chaos convection phenomena tornadoes, vortexes... 

The frequency of modulation il is now the main parameter, and we are able to switch the system of SHG between different dynamics by changing the value of il. To find the regions of where a chaotic motion occurs, we calculate a Lyapunov spectrum versus the knob parameter il. The first Lyapunov exponent A,j from the spectrum is of the greatest importance its sign determines the chaos occurrence. The maximal Lyapunov exponent Xj as a function of is presented for GCL in Fig. 6a and for BCL in Fig. 6b. We see that for some frequencies il the system behaves chaotically (A-i > 0) but orderly Ck < 0) for others. The system in the second case is much more damped than in the first case and consequently much more stable. By way of example, for = 0.9 the system of SHG becomes chaotic as illustrated in Fig. 7a, showing the evolution of second-harmonic and fundamental mode intensities. The phase point of the fundamental mode draws a chaotic attractor as seen in the phase portrait (Fig. 7b). However, the phase point loses its chaotic features and settles into a symmetric limit cycle if we change the frequency to = 1.1 as shown in Fig. 8b, while Fig. 8a shows a seven-period oscillation in intensities. To avoid transient effects, the evolution is plotted for 450 < < 500. 

A few words should be said about the difference between resonance and molecular vibrations. Although vibrations take place, they are oscillations about an equilibrium position determined by the structure of the resonance hybrid, and they should not be confused with the resonance among the contributing forms. The molecule does not resonate or vibrate" from one canonical structure to another. In this sense the term resonance is unfortunate because it has caused unnecessary confusion by invoking a picture of vibration. The term arises from a mathematical analogy between the molecule and the classical phenomenon of resonance between coupled pendulums, or other mechanical systems. 

At the root of these behaviors is the phenomenon of synchronization of a macroscopic oscillating system. This topic has been discussed occasionally in the literature but has rarely been explicitly treated. In principle there are several stages or hierarchical levels on which oscillations can occur (1) the single-crystal plane, (2) the catalytically active metal crystallite, (3) the catalyst pellet, (4) arrangements of several pellets in one layer of a flow reactor or a CSTR, and finally (5) the catalytic packed-bed reactor (327). On each of these levels, different types of oscillations may exist, but to become observable on the next level oscillations on the respective sublevels must be synchronized. For example, if oscillations of the CO/O2 reaction on a Pt(lOO) face of a Pt crystallite supported on a pelletized support material in a packed-bed reactor occur, the reaction on the (100) facet as a whole must oscillate in synchrony, other (100) facets of the crystallite have to synchronize, other crystallites in the pellet must couple to the first crystallite, and, finally, all pellets in one layer of the bed must display oscillations in synchrony. If the synchronization on one of these levels fails, different oscillators will superimpose and their effects will cancel. One would then only observe a possible increased level of noise in the measured conversion. On the other hand, if synchronization occurs independently over several regions of a system, then it might exhibit apparently chaotic behavior caused by incomplete coupling. 

Oscillating systems can interact with interesting results. Much like waves can cancel one another out, two oscillating systems can interact to produce a steady state of oscillation. On the other hand, the joining of two systems already at steady state can cause rhythmogenesis in which the two systems will depart from the steady state. 

So far, all of our examples of strange attractors have come from autonomous systems, in which the governing equations have no explicit time-dependence. As soon as we consider forced oscillators and other nonautonomous systems, strange attractors start turning up everywhere. That is why we have ignored driven systems until now—we simply didn t have the tools to deal with them. 

Although the earlier models by Sel kov exhibit only one limit cycle, later models have multiple solutions, both oscillatory and nonoscillatory. These models are quite interesting for the reason that these are the earliest models in biochemical oscillations giving clear evidence for more complex solutions than a simple limit cycle as has been the case with almost all other chemical systems except those in CSTR models. 

In Chapter I we found that curvilinear coordinates, such as spherical polar coordinates, are more suitable than Cartesian coordinates for the solution of many problems of classical mechanics. In the applications of wave mechanics, also, it is very frequently necessary to use different kinds of coordinates. In Sections 13 and 15 we have discussed two different systems, the free particle and the three-dimensional harmonic oscillator, whose wave equations are separable in Cartesian coordinates. Most problems cannot be treated in this manner, however, since it is usually found to be impossible to separate the equation into three parts, each of which is a function of one Cartesian coordinate only. In such cases there may exist other coordinate systems in terms of which the wave equation is separable, so that by first transforming the differential equation into the proper... 

Development of nuclear polarizations in the spin-correlated pairs or biradicals Because Equation (6) couples the nuclear spin motion and the electron-spin motion, not only the electron-spin state of each pair oscillates but also the nuclear spin state. Over the ensemble, however, the oscillation is not symmetrical because flip-flop fransifions are only possible for one-half of the pairs. Consider, for example, an ensemble of biradicals with one proton, and let the biradicals be bom in state F i). Taking into accoimt also the nuclear spin, one-half of fhe biradicals are fhus born in state T ia) and the other half in state T ijS). The latter cannot undergo flip-flop transitions to the singlet state, so have to remain in the state they were bom. The others oscillate between T ia) and I SjS). If a fracfion n of fhem has reached the singlet state, the total number of biradicals wifh nuclear spin a) is l-n)/2, and the total number of biradicals wifh spin jS) is n/2+1/2. The difference between the number of molecules wifh nuclear spin a) and jS) is fhus -n, in other words the system oscillates between zero polarization (n = 0) and complete polarization of one sort n = -1,... 

The situation is quite different with S-T -type CIDNP because nuclear spins are flipped in that case. Owing to the coupling of nuclear spin motion and electron spin motion, not only the electron spin state oscillates in such a system but also the nuclear spin state. Since, however, one-half of the pairs or biradicals cannot participate in this because their nuclear spin state does not allow an electron-nuclear flip-flop transition, the oscillation is not symmetrical. Its turning points are zero nuclear spin polarization and 100% nuclear spin polarization of one sign only. In contrast, the distribution of nuclear spin polarizations between singlet and triplet members of the ensemble is symmetrical. As an example, consider an ensemble of biradicals, where each biradical contains a single proton. Let the ensemble be created in the state T >, and without initial nuclear spin polarization. Half of the pairs, namely those that have nuclear spin /J>, cannot undergo flip-flop transitions. The others oscillate between T a> and S/3>. When all of those happen to be in S/ >, every nuclear spin of the triplet biradicals and every... 

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