Stable oscillations

At frequencies higher than this, a solid-state counter must be used. This is based on a stable oscillator and, in effect, counts the pulses generated during one cycle of the supply frequency. The range and accuracy of the instrument depends on the master oscillator frequency, but units capable of use over the whole range up to 600 MHz... 

It was observed that with a linear circuit and in the absence of any source of energy (except probably the residual charges in condensers) the circuit becomes self-excited and builds up the voltage indefinitely until the insulation is punctured, which is in accordance with (6-138). In the second experiment these physicists inserted a nonlinear resistor in series with the circuit and obtained a stable oscillation with fixed amplitude and phase, as follows from the analysis of the differential equation (6-127). 

If the roots are pure imaginary numbers, the form of the response is purely oscillatory, and the magnitude will neither increase nor decay. The response, thus, remains in the neighbourhood of the steady-state solution and forms stable oscillations or limit cycles. 

Figure 3. Phase portrait of the noiseless dynamics (43) corresponding to the linear Langevin equation (15) (a) in the unstable reactive degree of freedom, (b) in a stable oscillating bath mode, and (c) in an overdamped bath mode. (From Ref. 37.)...

When an electromagnetic wave interacts with resonators, the effect of quantization of all possible stationary stable oscillating amplitudes arises without the requirement of any specifically organized conditions (like the inhomogeneous action of external harmonic force). 

Without the external field, two stable oscillations with frequencies 0). and anparameter variations the sys tem can be driven from one stable oscillation to the other one, exhibiting a threshold and excitability. Furthermore, the external field can lead to a complete collapse of the small amplitude oscillation. The closer the external frequency X is to the internal one, Wj, the smaller is the critical field strength F, which is necessary for the breakdown of the small oscillation and the subsequent transition to the other one. For F >F, the system can only exist in the large amplitude state. Tfiis°state is stable with respect to F, but there exists a region, where partial quenching of the oscillation can occur. 

CN Chap. 8, Sect. 8.2. Has an accuracy of 0(6T2, H2), and is unconditionally but A-stable. Oscillates, however, with initial transients. With large A, these oscillations persist over many steps, rendering CN useless. The oscillations can be damped (Sect. 8.5.1) but this defeats to some extent the simplicity of the method. For LSV or chronopotentiometry simulations, however, where no sharp initial transients occur, this might be a good method. 

Example 11 The simplest system with stable oscillation the Lotka Volterra model... 

The simplest example of the kinetic scheme with stable oscillation around a stationary state (the "center" situation by Lyapunov) is the autocatalytic Lotka-Volterra scheme with two intermediates. The scheme was first described in the beginning of the past century. Consider the stability of the stationary state of the stepwise process... 

Although it may be clear how the amplifier and AW device produce a stable oscillator, the initial source of the signal that starts the oscillation is not as obvious. The answer lies in ever-present noise, small in ampiitude though it may be, that is invariably picked up and amplified by the am-plifier/AW device combination. 

Thus, perturbation measurements are best made when (1) highest accuracy is not necessary and (2) a substantial perturbation only to the region between IDTs is anticipated or (3) such a large perturbation is anticipated that it would be difficult to differentiate between a mode hop and the perturbation itself (e.g., the transition from a dry surface to one covered with liquid) or (4) such a large change in insertion loss is anticipated over the course of the measurement (e.g., a viscoelastic transition in a polymer layer) that it is not possible to set up and maintain a stable oscillator circuit. 

The chemical reaction is assumed to proceed when the molecule passes irreversibly from a region of configuration space identified as the reactant to a region identified as the product through a saddle point in the multidimensional potential surface. It is assumed that this saddle point is characterized by a local maximum of the potential along one degree of freedom (the reaction coordinate), while the other intramolecular degrees of freedom maintain stable oscillations about their local minima. This saddle point constitutes the potential barrier to the reaction. 

Under the appropriate conditions, the acoustic force on a bubble can be used to balance against its buoyancy, holding the bubble stable in the liquid by acoustic levitation. This permits examination of the dynamic characteristics of the bubble in considerable detail, from both a theoretical and an experimental perspective. Such a bubble is typically quite small compared to an acoustic wavelength (e.g., at 20 kHz, the resonance size is approximately 150 pm). For rather specialized but easily obtainable conditions, a single, stable, oscillating gas bubble can be forced into such large-amplitude pulsations that it produces sonoluminescence emissions on each (and every) acoustic cycle. Such SBSL is outside the scope of this chapter. 

Two views of the dynamics of ecosystems. In A, it appears that in some instances the system returns to a control state or is in a stable oscillation. Looking at the same system form the top (B) indicates that the systems are moving in quite different directions. 

The results demonstrate the relationship between the phase angle and energy dissipation. From a physics point of view, the action of attractive and repulsive, nonlinear tip-surface interactions contribute to the coexistence of two stable oscillation states in AM-AFM. The jump in the amplitude curve corresponds to transition from the attractive to the repulsive regime, near point b, where the phase changes sign. This will later be important for interpretations of phase images. 

The second important bifurcation that is connected with a stability change in a stationary state is the /fop/bifurcation. At a Hopf bifurcation, the real parts of two conjugate complex eigenvalues of J vanish, and as Hopf s theorem ensures, a periodic orbit or limit cycle is bom. A limit cycle is a closed loop in phase space toward which neighboring points (of the kinetic representation) are attracted or from which they are repelled. If all neighboring points are attracted to the limit cycle, it is stable otherwise it is unstable (see Ref. 57). The periodic orbit emerging from a Hopf bifurcation can be stable or unstable and the existence of a Hopf bifurcation cannot be deduced from the mere fact that a system exhibits oscillatory behavior. Still, in a system with a sufficient number of parameters, the presence or absence of a Hopf bifurcation is indicative of the presence or absence of stable oscillations. 

Figure 22. (a) Calculated cyclic voltammograms and stationary I/U curve for the formic acid oxidation model [Eq. (15)]. The anodic and cathodic scans are indicated by arrows. The dashed line shows the portion of the stationary state curve that corresponds to unstable steady states. The triangle at U = 0.6 V marks the location of the Hopf bifurcation, (b) Calculated one-parameter bifurcation diagram of the formic acid model steady-state coverage of OH, 6 oH. vs. applied voltage U. Solid line indicates a stable steady state (SS), the dashed line an unstable steady state, and the dot-dashed line shows the maximum amplitude of stable oscillations. (Reprinted with permission from P. Strasser, M. Eiswirth and G. Ertl, J. Chem. Phys. 107, 991-1003, 1997. Copyright 1997 American Institute of Physics.)... 

The application of oscillator circuits as sensor interface for QCM is the most common method. Since a quartz crystal is a resonant element, stable oscillation can be excited by quite simple circuits. They deliver a frequency analog output signal, which can be easily processed in digital systems. Two oscillation conditions can be formulated assuming approximately linear behavior and not considering the pre-oscillation process ... 

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