Coupled-oscillator

The concept of coupled oscillators is important from the viewpoint of understanding oscillatory phenomena in biochemical systems where the basic question relates to the type of dynamic behaviour when two are more such systems are coupled together. The concept of coupling is quite relevant in the context of spatial dissipative structures in cases where chemical oscillators are coupled through diffusion. 

On the basis of physical properties, the coupled oscillators can be divided in three categories  

A single oscillator may contain more than one set of reactions capable of oscillations. However, a coupled system exhibits more complex dynamical behaviour. The following types of complexities can occur in coupled oscillators  

For electrical coupling, large-area platinum electrodes are placed in each of the reactors to be coupled, and their circuit is completed by a wire between the electrodes and by an ion-bridge between the two reactors. Several phenomena such as entrainment, quasi-periodicity and chaos have been observed in such oscillations [55], 

Maselko for the first time [57] investigated chemical coupling in CSTR involving B-Z reactions with mixed substrates such as malonic acid-citric acid and malonic acid-oxalo-acetic acid. 

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Finally, semi-classical approaches to non-adiabatic dynamics have also been fomuilated and siiccessfLilly applied [167. 181]. In an especially transparent version of these approaches [167], one employs a mathematical trick which converts the non-adiabatic surfaces to a set of coupled oscillators the number of oscillators is the same as the number of electronic states. This mediod is also quite accurate, except drat the number of required trajectories grows with time, as in any semi-classical approach. 

When both vibrations have high frequencies, Wa, coq, the transition proceeds along the MEP (curve 1). In the opposite case of low frequencies, rUa.s the tunneling occurs in the barrier, lowered and reduced by the symmetrically coupled vibration q, so that the position of the antisymmetrically coupled oscillator shifts through a shorter distance, than that in the absence of coupling to qs (curve 2). The cases (0 (Oq, < (Oo, and Ws Wo, (Oq, characterized by combined trajectories (sudden limit for one vibration and adiabatic for the other) are also presented in this picture. 

Kaplunovsky and Weinstein [kaplu85j develop a field-theoretic formalism that treats the topology and dimension of the spacetime continuum as dynamically generated variables. Dimensionality is introduced out of the characteristic behavior of the energy spectrum of a system of a large number of coupled oscillators. 

The eigenvalue problem was introduced in Section 7.3, where its importance in quantum mechanics was stressed. It arises also in many classical applications involving coupled oscillators. The matrix treatment of the vibrations of polyatomic molecules provides the quantitative basis for the interpretation of their infrared and Raman spectra. This problem will be addressed tridre specifically in Chapter 9. 

A simple eigenvalue problem can be demonstrated by the example of two coupled oscillators. The system is illustrated in fug. 2. It should be compared with the classical harmonic oscillator that was treated in Section 5.2.2. Here also, the system will be assumed to be harmonic, namely, that both springs obey Hooke s law. The potential energy can then be written in the form... 

The relative displacements of the masses in the two normal modes of this coupled oscillator are shown to the right in Fig. 2. This method the form of the normal modes is particularly useful in the analysis of molec vibrations (see Chapter 9). 

Not only original signals (one or several) but additional combined signals (overtones, coupled oscillations, e.g. NIR) and latent signals in form of relations between original signals (differences, e.g. MS)... 

The formulation of the preceding section is very general. We are interested, however, in rotations and vibrations of polyatomic molecules. We therefore discuss now specific applications of the algebraic method beginning with the simple case of one-dimensional coupled oscillators, presented in Section 3.3 in the Schrodinger picture. In the algebraic theory, as mentioned, one associates to each coordinate, x, and related momentum, px = — iti d/dx, an algebra. For... 

For two coupled oscillators, the second possibility is chain (II) of Eq. (4.13). A dynamical symmetry corresponding to this route implies that the Hamiltonian operator contains only invariant operators of this chain,... 

The spectrum corresponding to Eq. (4.27) is shown in Figure 4.2. One can see that this represents the usual spectrum of two normal anharmonic coupled oscillators. 

In the preceding sections we have discussed the algebraic treatment of onedimensional coupled oscillators. We now present the general theory of two three-dimensional coupled rovibrators (van Roosmalen, Dieperink, and... 

We have already discussed in Section 4.5 the local-to-normal transition for two coupled oscillators. The situation is quite analogous for two coupled rovibrators. The local-to-normal transition can be described by combining the operators of the local chain with those of the normal chain. It is convenient to introduce the Majorana operator... 

Kellman, M. E. (1982), Group Theory of Coupled Oscillators Normal Modes as Symmetry Breaking, J. Chem. Phys. 76,4528. 

Voth, G. A. (1986), On the Relationship of Classical Resonances to the Quantum Mechanics of Coupled Oscillator Systems, J. Phys. Chem. 90, 3624. 

Fig. 13.21 shows another example of oscillatory burning of an RDX-AP composite propellant containing 0.40% A1 particles. The combustion pressure chosen for the burning was 4.5 MPa. The DC component trace indicates that the onset of the instability is 0.31 s after ignition, and that the instability lasts for 0.67 s. The pressure instability then suddenly ceases and the pressure returns to the designed pressure of 4.5 MPa. Close examination of the anomalous bandpass-filtered pressure traces reveals that the excited frequencies in the circular port are between 10 kHz and 30 kHz. The AC components below 10 kHz and above 30 kHz are not excited, as shown in Fig. 13.21. The frequency spectrum of the observed combustion instability is shown in Fig. 13.22. Here, the calculated frequency of the standing waves in the rocket motor is shown as a function of the inner diameter of the port and frequency. The sonic speed is assumed to be 1000 m s and I = 0.25 m. The most excited frequency is 25 kHz, followed by 18 kHz and 32 kHz. When the observed frequencies are compared with the calculated acoustic frequencies shown in Fig. 13.23, the dominant frequency is seen to be that of the first radial mode, with possible inclusion of the second and third tangential modes. The increased DC pressure between 0.31 s and 0.67 s is considered to be caused by a velocity-coupled oscillatory combustion. Such a velocity-coupled oscillation tends to induce erosive burning along the port surface. The maximum amplitude of the AC component pressure is 3.67 MPa between 20 kHz and 30 kHz. - ...

Denoting the electric and magnetic dipole transition moments of oscillators a and b as p, lUg and m, lUb, respectively, the rotational strength of the coupled oscillator is given by (34)... 

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