Curve oscillatory

Figure 14.6 Bifurcation curves in a model of the CDIMA reaction in the presence of an immobile complexing agent with c = 1.5. Solid line Hopf bifurcation curve for several values of cr steady state is stable above the curve, oscillatory state is stable below the curve. Dashed line Turing bifurcation curve homogeneous steady state is stable to inhomogeneous perturbations above, unstable below the curve. (Adapted from Lengyel and Epstein,...

The next problem to consider is how chaotic attractors evolve from tire steady state or oscillatory behaviour of chemical systems. There are, effectively, an infinite number of routes to chaos [25]. However, only some of tliese have been examined carefully. In tire simplest models tliey depend on a single control or bifurcation parameter. In more complicated models or in experimental systems, variations along a suitable curve in the control parameter space allow at least a partial observation of tliese well known routes. For chemical systems we describe period doubling, mixed-mode oscillations, intennittency, and tire quasi-periodic route to chaos. 

If a confined fluid is thermodynamically open to a bulk reservoir, its exposure to a shear strain generally gives rise to an apparent multiplicity of microstates all compatible with a unique macrostate of the fluid. To illustrate the associated problem, consider the normal stress which can be computed for various substrate separations in grand canonical ensemble Monte Carlo simulations. A typical curve, plotted in Fig. 16, shows the oscillatory decay discussed in Sec. IV A 2. Suppose that instead... 

Figure 3, Typical GPC traces—(a) sinusoidal feed Curve 1, steady-state unperturbed flow Curve 2, oscillatory steady-state, (b) Square-wave feed Curve 3, steady-state unperturbed flow Curve 4, oscillatory steady-state.

The surface forces apparatus (SEA) can measure the interaction forces between two surfaces through a liquid [10,11]. The SEA consists of two curved, molecularly smooth mica surfaces made from sheets with a thickness of a few micrometers. These sheets are glued to quartz cylindrical lenses ( 10-mm radius of curvature) and mounted with then-axes perpendicular to each other. The distance is measured by a Fabry-Perot optical technique using multiple beam interference fringes. The distance resolution is 1-2 A and the force sensitivity is about 10 nN. With the SEA many fundamental interactions between surfaces in aqueous solutions and nonaqueous liquids have been identified and quantified. These include the van der Waals and electrostatic double-layer forces, oscillatory forces, repulsive hydration forces, attractive hydrophobic forces, steric interactions involving polymeric systems, and capillary and adhesion forces. Although cleaved mica is the most commonly used substrate material in the SEA, it can also be coated with thin films of materials with different chemical and physical properties [12]. 

Fig. 2.8. Left oscillatory part of the reflectivity change of Bi (0001) surface at 8K (open circles). Fit to the double damped harmonic function (solid curve) shows that the Aig and Eg components (broken and dotted curves) are a sine and a cosine functions of time, respectively. Right pump polarization dependence of the amplitudes of coherent Aig and Eg phonons of Bi (0001). Adapted from [25]...

Fig. 2.18. (a) The even (upper panel) and odd (lower panel) SH responses of a 20 nm Gd(0001) film at 90 K using 815nm/35fs laser pulses. Transient reflectivity change is also displayed in the upper panel (solid black curve). The inset shows the experimental scheme with the magnetization oriented perpendicular to the plane of incidence, (b) The oscillatory part of the even and odd SH fields extracted from (a). The inset shows the corresponding FT spectra. From [59]... 

A passivating oxide is formed under sufficiently anodic potentials in HF, too. However, there are decisive differences to the case of alkaline and fluoride-free acidic electrolytes. For the latter electrolyte the steady-state current density prior to passivation is zero and it is below 1 mA cnT2 for alkaline ones, while it ranges from mA cm-2 to A cm-2 in HF. Furthermore, in HF silicon oxide formation does not lead to passivation, because the anodic oxide is readily etched in HF. This gives rise to an anodic I-V curve specific to HF, it shows two current maxima and two minima and an oscillatory regime, as for example shown in Fig. 4.7. 

Figure 4.72 presents the effect of the inhibition constant K. In this case the substrates are fed at constant concentrations, and the concentration of the external inhibitor in the feed stream changes according to the function described in Section 4.1.2. It can be seen that an oscillatory signal is obtained and its amplitude increases as K increases from 0.001 mM to 0.1 mM. Another increase in the value of K from 0.1 mM to 1 mM causes the opposite effect, and the amplitude decreases drastically. This relationship between the amplitude of the output signal and the value of K is presented in Figure 4.73, where a bell-shaped curve is observed. The amplitude of the input signal is 9 mM therefore, the system presented decreases this amplitude by 600 or more. This enables one to obtain very fine amplitudes that cannot be obtained by means of flow rates. 

Even though the bifurcation behavior exhibits a Z-shaped curve, it is more complicated due to the existence of the HB. For example, upon ignition, the system is expected to oscillate because no locally stable stationary solutions are found (an oscillatory ignition). Time-dependent simulations confirm the existence of self-sustained oscillations [7, 12]. The envelope of the oscillations (amplitude of H2 mole fraction) is shown in circles (a so-called continuation in periodic orbits). 

Figure 28. Spherical flaw growth caused by oscillatory displacement u = u sin wt for a polyurethane elastomer (5 c.p.s.). Curve generated by Equation 28...

The resultant pair potentials for sodium, magnesium, and aluminium are illustrated in Fig. 6.9 using Ashcroft empty-core pseudopotentials. We see that all three metals are characterized by a repulsive hard-core contribution, Q>i(R) (short-dashed curve), an attractive nearest-neighbour contribution, 2( ) (long-dashed curve), and an oscillatory long-range contribution, 3(R) (dotted curve). The appropriate values of the inter-atomic potential parameters A , oc , k , and k are listed in Table 6.4. We observe that the total pair potentials reflect the characteristic behaviour of the more accurate ab initio pair potentials in Fig. 6.7 that were evaluated using non-local pseudopotentials. We should note, however, that the values taken for the Ashcroft empty-core radii for Na, Mg, and Al, namely Rc = 1.66, 1.39, and... 

The link between the oscillatory behaviour of the structural energy curves and the moments of the local density of states can be made explicit by writing the bond order of a given bond as a many-atom expansion about that bond (Pettifor (1989)). Considering for simplicity the case of s orbitals, on a lattice where all sites are equivalent, the bond order can be expressed exactly (Aoki (1993)) as... 

NAD(P)H oxidase can also be activated by fluid shear stresses. This is one reason why branched and curved arteries tend to develop atherosclerotic plaques earlier than straight arteries. Using a spin probe to detect 02 , Hwang et al. have demonstrated that monolayer cultures of EC exposed to oscillatory (but not laminar) shear stresses produce the radical using NAD(P)H oxidase.292 A subsequent study showed that XO also responds to oscillatory shear stress.293 Other workers, using BMPO, have detected the flow-induced production of 02 by mitochondria.294... 

If, however, we actually integrate the reaction rate equations numerically using the rate constants in Table 1.1 we find that the system does not always stick to, or even stay close to, these pseudo-steady loci. The actual behaviour is shown in Fig. 1.10. There is a short initial period during which d and b grow from zero to their appropriate pseudo-steady values. After this the evolution of the intermediate concentrations is well approximated by (1.41) and (1.42), but only for a while. After a certain time, the system moves spontaneously away from the pseudo-steady curves and oscillatory behaviour develops. We may think of the. steady state as being unstable or, in some sense repulsive , during this period in contrast to its stability or attractiveness beforehand. Thus we have met a bifurcation to oscillatory responses . The oscillations... 

Let us return to Fig. 1.12(c), where there are multiple intersections of the reaction rate and flow curves R and L. The details are shown on a larger scale in Fig. 1.15. Can we make any comments about the stability of each of the stationary states corresponding to the different intersections What, indeed, do we mean by stability in this case We have already seen one sort of instability in 1.6, where the pseudo-steady-state evolution gave way to oscillatory behaviour. Here we ask a slightly different question (although the possibility of transition to oscillatory states will also arise as we elaborate on the model). If the system is sitting at a particular stationary state, what will be the effect of a very small perturbation Will the perturbation die away, so the system returns to the same stationary state, or will it grow, so the system moves to a different stationary state If the former situation holds, the stationary state is stable in the latter case it would be unstable. 

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