Amplitude, stability diagram

The study of the stability properties of the unique steady state admitted by eqns (5.1) permits us to establish the stability diagram of fig. 7.2 in which the dashed area C denotes the domain of instability of the steady state, where sustained oscillations of the limit cycle type occur. In domain B, the steady state is stable but excitable, as the system amplifies, in a pulsatory manner, a cAMP signal whose given amplitude exceeds a threshold. Everywhere else in the diagram the steady state is stable but nonexcitable. 

Fig. 7.2. Developmental path of the cAMP signaUing system in the parameter space formed by adenylate cyclase and phosphodiesterase activity. The stability diagram is established by linear stability analysis of the steady state admitted by the three-variable system (5.1) governing the dynamics of the allosteric model for cAMP signalling in D. discoideum (see section 5.2). In domain C sustained oscillations occur around an unstable steady state. In domain B, the steady state is stable but excitable as it amplifies in a pulsatile manner a suprathreshold perturbation of given amplitude. Outside these domains the steady state is stable and not excitable. The arrow crossing successively domains A, B and C represents the developmental path that the system should follow in that parameter space to account for the observed sequence of developmental transitions no relay relay oscillations (Goldbeter, 1980).

The stability diagram of the Mathieu equation in the plane of the parameters S and is called the Strutt-Ince diagram (van der Pol and Strutt 1928). This diagram can be transformed to the plane of the forcing amplitude r and the forcing frequency /= col2%. 

FIGURE 1. (a) Parametron semiconductor circuit with a one-to-one correspondence analogue of a dipole double layer. Inductance L, resistance R, and capacitors C, C V). (b) Stability diagram of behavior of the analog model (a) Q = cdL/R is a performance parameter of the circuit and yo is a function of Q, (Oq, 0 (the junction potential of the semiconductor diode), and the amplitude of sinusoidal voltage. Numbers 0, 1, 1, 2,... are dominant fre-... 

Figure 5. Stability diagram with curves of equal temperature amplitude (B = 15,( = 0,02075)...

Over the years, several methods of isolating ions in a QIT have been implemented. Methods include RF/DC isolation, forward and reverse RF resonance ejection isolation, " and various forms of TWF isolation. " The RF/DC isolation methods position the ion of interest near the boundaries of the stability diagram for isolation. In this case, the parameter is set to a nonzero value (see RF/DC isolation point in Figure 9.3b). The RF resonance ejection method sweeps the main RF amplitude and/or the resonance ejection frequency in both the forward and/or reverse directions to eject all but the ion of interest. This technique has been shown to yield high-resolution isolations and can be used to analyze multiply charged ions. ... 

In Eq. (9.8), V max is the maximum amplitude of the RF and z ject is the point on the stability diagram where the ions are resonantly ejected. By using a lower z-eject, however, the scan rate (Th s ) increases, but resolution is reduced. In the example above, the scan rate would be increased to 55,550 Th s . This rapid scan rate will also result in fewer points acquired across a mass peak unless a higher data acquisition rate is used (see Section 9.3.3.3, The Effects o/Scan Rate The Normal, Rapid, and Slow Scan Rates )- A solution to this problem would be to reduce the scan rate back to 5555 Th s and to limit the mass range to avoid long scan times. Mass range extension in the LQIT can be accomplished by applying the same methods as described above. 

Figure 8.10 Stability diagram for an infinitely long and perfect quadrupole in the dimensionless space of (a, q), where a and q are the Mathieu parameters given by equations (8.2.4) in the text. Ions that are within the enclosed region have finite (stable) trajectories, independent of their initial displacement and initial phase of the applied r.f. Ions that are outside this region have amplitude of their oscillations exponentially increasing and are thus unstable in the field - independent of their initial displacement and initial phase of the applied r.f. Also shown are the iso-beta lines ions of the same beta have same frequencies of trajectory oscillation in the field.

The amplitude of the periodic orbits is therefore determined by the linear stability with respect to perturbations transverse to the orbit. In this sense, the leading term in expression (2.13), obtained by setting C = 0, treats the dynamics transverse to the orbit at the level of the harmonic approximation. The nonlinear stability properties appear thus as anharmonic corrections to the dynamics transverse to the orbit. These anharmonicities contribute to the trace formula by corrections given in terms of series in powers of the Planck constant involving the coefficients C , which can be obtained as Feynman diagrams [14, 31]. 

This heuristic argument forms the basis of the Bode stability criterion(22,24) which states that a control system is unstable if its open-loop frequency response exhibits an AR greater than unity at the frequency for which the phase shift is —180°. This frequency is termed the cross-over frequency (coco) for reasons which become evident when using the Bode diagram (see Example 7.7). Thus if the open-loop AR is unity when i/r = —180°, then the closed-loop control system will oscillate with constant amplitude, i.e. it will be on the verge of instability. The greater the difference between the open-loop AR (< I) at coc and AR = 1, the more stable the closed-loop... 

The principles underlying these observations are summarized in the pha diagram of Fig. 12. The amplitude of oscillations is plotted as a function of the ratio of stabilizing and destabilizing agents (for details see [16]). When microtubules are stabilized (left part of diagram) one observes microtubules without oscillations. 

The time-domain methods enable a very realistic prediction of the behavior of the machine tool, since the effective kinematics of the machining process as well as nonlinear effects (e.g., tool jumping) are taken into account (Totis 2009). Adversely, such simulations cause a lot of computation time, due to the high complexity. Furthermore, the identification of the whole stability lobe diagram requires its complete simulation for each combination of process parameters. To determine the limit of stability, changes of the vibration response or of the force amplitudes are detected during each run of simulation. 

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