Forward bifurcation

It can be seen from Fig. 15(a) that the atom moves in a stick-slip way. In forward motion, for example, it is a stick phase from A to B during which the atom stays in a metastable state with little change in position as the support travels forward. Meanwhile, the lateral force gradually climbs up in the same period, leading to an accumulation of elastic energy, as illustrated in Fig. 15(fo). When reaching the point B where a saddle-node bifurcation appears, the metastable... 

The changes of lateral force F in forward and backward motions follow the curve 1 and 2, respectively. It can be observed that there is one saddle-node bifurcation for the repulsive pinning center, but two bifurcations for the attractive piiming center. This suggests that the interfacial instability results from different mechanisms. On one hand, the asperity suddenly looses contact as it slides over a repulsive pinning center, but in the attractive case, on the other hand, the... 

Clearly flow aligning behavior of the director is present and do increases linearly with the tilt angle, do. Above a threshold in the Spain rate, y 0.011, undulations in vorticity direction set in. In Fig. 14 the results of simulations for y 0.015 are shown. In Fig. 15 we have plotted the undulation amplitude obtained as a function of the shear rate. The dashed line indicates a square root behavior corresponding to a forward bifurcation near the onset of undulations. This is, indeed, what is expected, when a weakly nonlinear analysis based on the underlying macroscopic equations is performed [54], In Fig. 16 we have plotted an example for the dynamic behavior obtained from molecular dynamics simulations. It shows the time evolution after a step-type start for two shear rates below the onset of undulations. The two solid lines correspond to a fit to the data using the solutions of the averaged linearized form of (27). The shear approaches its stationary value for small tilt angle (implied by the use of the linearized equation) with a characteristic time scale t = fi/Bi. 

As usual in bifurcation theory, there are several other names for the bifurcations discussed here. The supercritical pitchfork is sometimes called a forward bifurcation, and is closely related to a continuous or second-order phase transition in sta-... 

In the following, we first discuss the situations where EC occurs as a primary forward bifurcation and where the standard model is directly apphcable (cases A and B). Then we discuss configurations where EC sets in as a secondary instability upon an already distorted Freedericksz ground state and compare it with experiments (cases C and D). Note that in this case the linear analysis based on the standard model already becomes numerically demanding. Finally, we address those combinations of parameters where a direct transition to EC is not very robust, since it is confined to a narrow Ca range around zero. For cases E and H this range may be accessible experimentally while for cases F and G it is rather a theoretical curiosity only. 

Measurements have shown that the prewavy pattern appears in a forward bifurcation [50]. Its threshold voltage Upw has a weak, nearly linear frequency dependence. It usually occurs at higher frequencies (see Fig. 1). Conductive normal rolls, dielectric rolls and the prewavy pattern may follow each other with increasing / (dielectric rolls may be skipped in compounds with higher conductivity). Near the crossover frequency /c, the conductive (or dielectric) rolls can coexist with the prewavy pattern resulting in the defect-free chevron structure [51]. 

This conclusion is not quite as straightforward for other kinetic terms. For example, if F(p) = p(l - p) p -a),then Q(p) = p (2p —1—a), which does not have a definite sign on [0, L]. If (2(p) > 0 on [0, L], the nonuniform steady state is stable. If <2(P) < 0 on [0, L], the nonuniform steady state is unstable. If Q( ) changes sign on [0, L], no conclusion can be drawn and one has to resort to other tools. The bifurcation diagram that emerges for the logistic case is that of a simple forward or supercritical bifurcation at as illustrated in Fig. 9.2. Below the trivial steady state is stable and the population dies out. Above L, the nonuniform steady state is stable, the trivial state is unstable, and the population persists or survives. 

In Fig. 9.4 we plot the corresponding bifurcation diagram. A forward or supercritical bifurcation occurs at T = L. We depict with symbols the values obtained by integrating (9.1) numerically using an explicit finite difference method with a... 

There has been only one investigation of flexodomains in the weakly nonlinear regime for U > Vein the DC case. Based on a clever variational ansatz for the director distortion it has been demonstrated that the director amplitudes grow continuously as (U — (forward bifurcation). As a... 

A multilayered feed forward neural network with input, hidden and output layer is chosen. The choice follows recommendation of Hurtado Alvares (2001), which argue that radial basic functions (RBF) networks are not suitable for bifurcation problems. 

Anatomic Consideration The internal iliac arteries, the blood supply to the viscera of the true pelvis, are readily approached after femoral arterial access. The ipsilateral internal iliac artery is usually catheterized with a reverse curve catheter configuration and the contralateral internal iliac artery is usually accessed following passage over the aortic bifurcation with a forward seeking cobra catheter. On rare occasions because of atherosclerotic stenosis or occlusion of one femoral artery, two catheters (4-5 F) can be... 

As with multiple shooting approaches, simultaneous approaches can deal with instabilities that occur for a range of inputs. Because they can be seen as extensions of robust boundary valne solvers, they are able to pin down unstable modes (or increasing modes in the forward direction). This property has important advantages for problems that include transitions to unstable points and optimization of systems with limit cycles and bifurcations. 

The exciting issue of steady-state multiplicity has attracted the attention of many researchers. First the focus was on exothermic reactions in continuous stirred tanks, and later on catalyst pellets and dispersed flow reactors as well as on multiplicity originating from complex isothermal kinetics. Nonisothermal catalyst pellets can exhibit steady-state multiplicity for exothermic reactions, as was demonstrated by P.B. Weitz and J.S. Hicks in a classical paper in the Chemical Engineering Science in 1962. The topic of multiplicity and oscillations has been put forward by many researchers such as D. Luss, V. Balakotaiah, V. Hlavacek, M. Marek, M. Kubicek, and R. Schmitz. Bifurcation theory has proved to be very useful in the search for parametric domains where multiple steady states might appear. Moreover, steady-state multiplicity has been confirmed experimentally, one of the classical papers being that of A. Vejtassa and R.A. Schmitz in the AIChE Journal in 1970, where the multiple steady states of a CSTR with an exothermic reaction were elegantly illustrated. 

Fig. 11.3.6. Bifurcations of a saddle-node limit cycle in a plane for the case I2 < 0. This is the same as in Fig. 11.3.5 up to a change in /i —V /x. The feature of the forward route is the appearance of a saddle-node cycle from condensation of trajectories.

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