Oscillatory waveform

Other complex oscillatory waveforms include bursting , in which large-amplitude oscillations are interspersed with periods of non-oscillatory evolution, or by small oscillations (Fig. 1.19). 

Fig. 2.6. Variation of oscillatory waveform with precursor reactant concentration (a) stationary-state locus for intermediate A, a(p), showing region of instability for p < p < p indicating the magnitude of the oscillations (b) oscillations in a and b and the corresponding limit cycle for p = 0.005 (c) oscillations in a and h and the corresponding limit cycle for p = 0.010 (d) oscillations in a and b and the corresponding limit cycle for p = 0.0195.

The examples we saw are for L C circuits supplied from a direct current source. What happens when an L C circuit is excited by an alternating current source Once again, oscillatory response will be present. The oscillatory waveform superimposes on the fundamental waveform until the damping forces sufficiently attenuate the oscillations. At this point, the system returns to normal operation. In a power system characterized by low resistance and high values of L and C, the effects would be more damaging than if the system were to have high resistance and low L and C because the natural frequencies are high when the values of L and C are low. The... 

Synchronization through the gas phase was also assumed to occur by Tsai et al. (82) for CO oxidation over polycrystalline Pt. At 760 Torr the observed oscillations exhibited complex waveforms. When the pressure was lowered to 10 Torr, communication increased between oscillating patches of the catalyst due to higher gas-phase diffusivity and the oscillatory waveforms become simpler. 

Fig. 5.24. Variation of oscillatory waveform in vicinity of boundary between oscillatory and steady ignition showing characteristic nature of a supercritical Hopf bifurcation. (Reprinted with permission from reference [33], Royal Society of Chemistry.)...

In order to model the oscillatory waveform and to predict the p-T locus for the (Hopf) bifurcation from oscillatory ignition to steady flame accurately, it is in fact necessary to include more reaction steps. Johnson et al. [45] examined the 35 reaction Baldwin-Walker scheme and obtained a number of reduced mechanisms from this in order to identify a minimal model capable of semi-quantitative p-T limit prediction and also of producing the complex, mixed-mode waveforms observed experimentally. The minimal scheme depends on the rate coefficient data used, with an updated set beyond that used by Chinnick et al. allowing reduction to a 10-step scheme. It is of particular interest, however, that not even the 35 reaction mechanism can predict complex oscillations unless the non-isothermal character of the reaction is included explicitly. (In computer integrations it is easy to examine the isothermal system by setting the reaction enthalpies equal to zero this allows us, in effect, to examine the behaviour supported by the chemical feedback processes in this system in isolation... 

Fig. 5.34. Traverse through region of complex ignition in Fig 5.33 for p = 40Torr (upper trace) and p = 20Torr (lower trace) showing evolution of complex oscillatory waveform. (Reprinted with permission from reference [67], American Institute of Physics.)...

Among the variously proposed mechanisms of exercise hyperpnea, the PCO2 oscillation hypothesis of Yamamoto [1962] has received widespread attention. According to this hypothesis, the controller may be responsive not only to the mean value of chemical feedback but also to its oscillatory waveform which is induced by the tidal rhythm of respiration. This hypothesis is supported by the experimental finding that alterations of the temporal relationship of the PaC02 waveform could profoundly modulate the exercise hyperpnea response (Poon, 1992b]. 

Figure 9 displays the transition through families of hybrid oscillatory waveforms as the vessel temperature is raised rapidly by 4 K and then allowed to diminish slowly. Temperature excesses and oxygen consumption are displayed. 

The broad principles underlying the existence of stationary states (both nodal and focal stabilities), simple oscillatory states and multiple stabilities, involving competitive chain branching and non-branching modes and the interaction with heat release and heat loss, are clear. The more complex oscillatory waveforms do not emerge naturally from this simple recipe, and greater chemical complexity, not yet taken into account, is probably their cause. 

Close to the upper end of the range of instability, the oscillations have small amplitude and a short period near p, the waveform is close to sinusoidal. As p is decreased the excursions increase in amplitude, quite quickly, attaining a maximum at p x 0.015 mol dm- 3 with the particular values of the rate constants used here. The period is now longer and the waveform less symmetric. At yet lower reactant concentrations, the amplitude decreases slightly the period continues to increase smoothly as p decreases over most of the oscillatory range, and the oscillations become more and more sawtooth in form. Finally, extremely close to p, the oscillatory amplitude and the period decrease rapidly again. The amplitude tends to zero, although the period remains finite. 

With this identification, the stable stationary-state behaviour (found for the cubic model with 1 < A < 4) corresponds to oscillations for which each amplitude is exactly the same as the previous one, i.e. to period-1 oscillatory behaviour. The first bifurcation (A = 4 above) would then give an oscillation with one large and one smaller peak, i.e. a period-2 waveform. The period doubling then continues in the same general way as described above. The B-Z reaction (chapter 14) shows a very convincing sequence, reproducing the Feigenbaum number within experimental error. 

Bates 1984 Fredrickson and Larson 1987 Fredrickson andFIelfand 1988). The relaxation of these fluctuations involves collective motion of many molecules, and thus it is slower than the relaxation time of individual molecules. In small-amplitude oscillatory shearing, the fluctuation waveform is deformed, producing a slowly relaxing stress. Presumably, this accounts for (a) the anomalous contribution to G and (b) a similar, but smaller, contribution to G" (Rosedale and Bates 1990 Jin and Lodge 1997). (Similar anomalies are observed in polymer blends.) An asymmetric version of this PEP-PEE polymer that forms cylindrical domains shows an even larger low-frequency anomaly (Almdal et al. 1992). 

Unusual stress waveforms are observed during oscillatory deformation on such materials when the strain amplitude is much greater than the yield strain. Watanabe and Kotaka (1984) plotted large-amplitude oscillatory shear data in the form of Lissajous figures— that is, plots of periodic stress versus periodic strain. In the nonlinear regime, above the yield strain, unusual rhombic Lissajous figures were obtained at low frequencies, and bent ellipses were obtained at higher frequencies. Qualitatively, the waveforms are similar to those depicted in Fig. 13-23. 

Fig. 3. Typical ERG waveforms obtained from the rat. These signals were collected for a range of light intensities (shown on the right-hand side). Note how at low light levels, a (positive) slow, rod driven b-wave is seen. This grows and speeds up as a result of cone contribution and develops oscillatory potentials (OPs). The negative trough is called the a-wave and rcllects photoreceptor activation see Fig. 2).

Their selection should be carefully conducted, since they function as pure weighting factors and therefore they can strongly subdue dispersion and dissipation errors. Also, correction functions df co, At) and LA() of (3.71) and (3.74) are selected to lessen grid discrepancies and certify the proper transition from the continuous physical space to the discretized domain. In fact, their arguments have a substantial contribution in the method s wideband profile and hence involve an in-depth examination. More specifically, by considering the excitation frequency content and duration, they subdue oscillatory or spurious modes that corrupt the final waveform envelope. Probing kf, LA() analysis indicates that its argument should opt... 

The individual components of an applied oscillatory torque of constant amplitude are illustrated in Figure 2.10, and the resulting complex torque signal, obtained from the superposition of these, is shown in Figure 2.11. This non-sinusoidal waveform for the applied torque results in an angular... 

Sharp cutoff filters should be avoided in biopotential measurements where the bioelectric waveform shape is of interest Filtering can greatly distort waveforms where waveform frequencies are near the filter breakpoints. Phase and amplitude distoitions are more severe with higher-order sharp-cutoff filters. Filters such as the Elliptic and the Tchebyscheff exhibit drastic phase distortion that can seriously distort bioelectric waveforms. Worse still for biopotential measurements, these filters have a tendency to ring or overshoot in an oscillatory way with transient events. The result can be addition of features in the bioelectric waveform that are not really present in the raw signal, time delays of parts of the waveforms, and inversion of phase of the waveform peaks. Figure 17.34 shows that the sharp cutoff of a fifth-order elliptical filter applied to an ECG waveform produces a dramatically distorted waveform shape. 

Time-domain methods are often used to characterize linear circuits, and can also be used to describe resonance. When an electrical circuit exhibits an undamped oscillatory or slightly damped behavior it is said to be in resonance, and the waveforms of the voltages and currents in the circuit can oscillate indefinitely. 

This can be verified by substituting the expression for v t) into the differential equation model and performing the indicated operations. The fact that v t) can be shown to have this form indicates that it is possible for this circuit to sustain oscillatory voltage and current waveforms indefinitely. When the parametric expression for v(t) is substituted into the differential equation model the value of co that is compatible with the solution of the equation is revealed to be ct) = l/.-/(LC). This is an example of the important fact that the frequency at which an electrical circuit exhibits resonance is determined by the physical value of its components. The remaining parameters ofv(t), K, and (p are determined by the initial energy stored in the circuit (i.e., the boundary conditions for the solution to the differential equation model of the behavior of the circuit s voltage). 

Gibbs phenomenon Refers to an oscillatory behavior in the convergence of the Fourier transform or series in the vicinity of a discontinuity, typically observed in the reconstruction of a discontinuous X (t). Formally stated, the Fourier transform does converge uniformly at a discontinuity, but rather, converges to the average value of the waveform in the neighborhood of the discontinuity. 

A number of studies in the field of EEG synchronization usethe coherence measure. However, it is argued that coherence cannot be regarded as a specific measure of synchronization [30], [31], [32]. As we know, coherence does not separatethe effects of covariance of the amplitude waveforms and of the phases of two oscillatory signals. Since the core of the synchronization is the adjustment of phases and not of amplitudes, it should be detected by a measure neglecting amplitude variations. 

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