Growth of oscillations

The size and period of the oscillations, or of the corresponding limit cycle, varies with the dimensionless reactant concentration pi. We may determine this dependence in a similar way to that used in 2.5. Close to the Hopf bifurcation points we can in fact determine the growth analytically, but in general we must employ numerical computation. For now we will merely present the basic result for the present model. The qualitative pattern of response is the same for all values of ku g. 

As the dimensionless concentration of the reactant decreases so that pi just passes through the upper Hopf bifurcation point pi in Fig. 3.8, so a stable limit cycle appears in the phase plane to surround what is now an unstable stationary state. Exactly at the bifurcation point, the limit cycle has zero size. The corresponding oscillations have zero amplitude but are born with a finite period. The limit cycle and the amplitude grow smoothly as pi is decreased. Just below the bifurcation, the oscillations are essentially sinusoidal. The amplitude continues to increase, as does the period, as pi decreases further, but eventually attains a maximum somewhere within the range pi% pi pi. As pi approaches the lower bifurcation point /zf from above, the oscillations decrease in size and period. The amplitude falls to zero at this lower bifurcation point, but the period remains non-zero. 

There are no unstable limit cycles in this model, and the oscillatory solution born at one bifurcation point exists over the whole range of stationary-state instability, disappearing again at the other Hopf bifurcation. Both bifurcations have the same character (stable limit cycle emerging from zero amplitude), although they are mirror images, and are called supercritical Hopf bifurcations. 

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To learn more about the details of the extent, the onset, and the growth of oscillations we are able to perform a complete Hopf analysis on the present model. We do this in detail, as an introduction to the full application of the Hopf technique, in the next chapter. The main results are, however, presented in the next section. 

In the next chapter we will derive additional aspects about the birth and growth of oscillations in this system. The natural frequency a>0 of the oscillations as they are born at one or other of the Hopf points can be evaluated explicitly and is related to the stationary-state dimensionless temperature rise ... 

Hopf bifurcation analysis with Arrhenius model birth and growth of oscillations... 

Growth of oscillations for heterogeneous catalysis model with poisoning A = amplitude, Tp = period... 

In hydrodynamic voltammetry current is measured as a function of the potential applied to a solid working electrode. The same potential profiles used for polarography, such as a linear scan or a differential pulse, are used in hydrodynamic voltammetry. The resulting voltammograms are identical to those for polarography, except for the lack of current oscillations resulting from the growth of the mercury drops. Because hydrodynamic voltammetry is not limited to Hg electrodes, it is useful for the analysis of analytes that are reduced or oxidized at more positive potentials. 

When two layers of the substance are displaced relative to one another, the nuclei of phase A, located between them, can be regarded as kind of a roller about which oscillations are executed. - - - when the two layers of phase AB are displaced relative to one another, they transport past the nucleus, in its immediate vicinity, a multiplicity of atoms of both kinds. - - - it follows that all the atoms A passing in the immediate vicinity of the nucieus have sufficient time to combine with the latter and this in fact may be the mechanism of the growth of the nuclei of the new phase. ... 

In the process of MBE, the surface structure can be investigated by reflected high energy electron diffraction (RHEED). During MBE growth, one often observes an oscillation in the intensity of the specular reflected beam as a function of time. This is interpreted to be due to the layer-by-layer growth of a two-dimensional island. 

A. Daniluk, P. Mazurek, K. Paprocki, P. Mikolajczak. RHEED intensity oscillations observed during the growth of YSi2 t on Si(lll) substrates. Surf SciSQl 226, 991. 

The bubble formed in stable cavitation contains gas (and very small amount of vapor) at ultrasonic intensity in the range of 1-3 W/cm2. Stable cavitation involves formation of smaller bubbles with non linear oscillations over many acoustic cycles. The typical bubble dynamics profile for the case of stable cavitation has been shown in Fig. 2.3. The phenomenon of growth of bubbles in stable cavitation is due to rectified diffusion [4] where, influx of gas during the rarefaction is higher than the flux of gas going out during compression. The temperature and pressure generated in this type of cavitation is lower as compared to transient cavitation and can be estimated as ... 

With decreasing temperature, the density oscillation becomes very pronounced and grows into a deeper melt region. At 300 K, for example, we can see at least 5 layers after 1.28 ns. Within the layers, as will be shown later, definite order in chain orientation and chain packing is observed suggesting the growth of polymer crystals. 

Polarography is the classical name for LSV with a DME. With DME as the working electrode, the surface area increases until the drop falls off. This process produces an oscillating current synchronized with the growth of the Hg-drop. A typical polarogram is shown in Fig. 18b. 10a. The plateau current (limiting diffusion current as discussed earlier) is given by the Ilkovic equation... 

Smog chamber studies have documented similar aerosol growth mechanisms. For example, in the photochemical oxidation of dimethyl sulfide, the formation and growth of particles in an initially particle-free system was observed. However, if seed particles with 34-nm mean size were present, an oscillation in the... 

Fig. 3.8. Representation of the onset, growth, and death of oscillations in the isothermal autocatalytic model as /z varies for reaction with the uncatalysed step included, showing emergence of the stable limit cycle at and its disappearance at n. ...

We have now seen how local stability analysis can give us useful information about any given state in terms of the experimental conditions (i.e. in terms of the parameters p and ku for the present isothermal autocatalytic model). The methods are powerful and for low-dimensional systems their application is not difficult. In particular we can recognize the range of conditions over which damped oscillatory behaviour or even sustained oscillations might be observed. The Hopf bifurcation condition, in terms of the eigenvalues k2 and k2, enabled us to locate the onset or death of oscillatory behaviour. Some comments have been made about the stability and growth of the oscillations, but the details of this part of the analysis will have to wait until the next chapter. 

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