Stable equilibrium point

Property 1. Consider an exothermic continuous stirred-tank reactor with temperature dependence Arrenhius-type, there is a stable equilibrium point such that, under the isothermic operation (i.e., as reactor temperature X2 is constant). 

Of the two stable equilibrium points, D may be further classified as absolutely stable (deepest minimum) and A as metastable (local minimum). The unstable point B is also called a transition state. Critical point C is a marginally stable state that may be further classified as stable or unstable according to the sign of the lowest nonvanishing derivative. 

These results are supported by the standard stability analysis of Figure 11.2, where A is set to 0.1 and y = 2 (y = k ). The eigenvalues computed by (11.6) are plotted as functions of y. In this figure, unstable and stable equilibrium points are clearly separated by an interval, [0.1974 0.2790], where eigenvalues are complex, leading to a stable focus. With increasing A, this interval becomes narrower and for A > 0.65, the eigenvalues have only real parts. 

Figure 11.15 The state space of the dimensionless set-point and temperature variables x (t) and y(t), respectively. Solid and dashed lines correspond to the low and high doses, respectively. ( ) represents the stable equilibrium point.

Marginally stable equilibrium points are such that d V/dq = 0. The stability is then determined by the nonlinearity of the force —dV/dq. In the 1-DOF case, this is easily extended to the general Hamiltonian (8). An equilibrium point is such that dH/dp = dH/dq = 0. The linear stability proceeds as follows. 

Other optical-based levitation methods use two or more laser beams to generate only one stable equilibrium point below the beam-crossing point [85]. Typical applications include 3D cellular and intracellular micromanipulation [86], and cell sorting [87]. 

Recent papers by Othmer ( 7) and Caram and Scriven (8) have pointed out that uniqueness is characteristic of ideal systems whereas for non-ideal systems a solution may occur at the global minimum (most stable equilibrium point) but it also may occur at a nonunique local minimum. For applications in aquatic chemistry the problem of nonuniqueness is particularly important in the interpretation of solid precipitation and dissolution processes. 

Fig. 5.4. Entropy (S) - Volume (Vi) - Energy (U) plot for the system in Figure 5.2. A three-dimensional version of Figure 5.3 (a) shows constant-U contours. Total volume V is also constant, (b) constant U, V contour and constant S, V contour through the same stable equilibrium point A. Point B represents a metastable equilibrium.

Is there a unique, asymptotically stable equilibrium point of the system (4.6) ... 

For the case of equal mutation rate (i.e if b j is independent of j) the system is Shahshahani gradient system. For any other bij Wy can be found such that the Shahshahani gradient property fails and periodic behaviour occurs via Hopf bifurcation. In terms of genetics this means that the frequency distribution of the alleles does not converge to a stable equilibrium point, but may exhibit oscillations. Even the occurrence of chaos seems probable in slightly... 

Notice that, if we consider two trajectories starting from two points that are close to each other in phase space in the vicinity of the stable equilibrium point, the difference between the two trajectories will always remain of the same order of magnitude as the initial difference the two trajectories will remain about as close to each other as their initial points are. 

The system length is supposed to be infinite. Without diffusion, the system is assumed to admit one or more stable equilibrium points but no stable oscillations. Furthermore, these equilibrium states are supposed to remain stable to nonuniform fluctuations. As is well known, it sometimes happens that steadily traveling nonlinear waves arise in such non-oscillatory media. (For a special model, see Sect. 7.3.) Two types of waves are possible, which are schematically... 

The averaged system has an exponentially stable equilibrium point z = 0. 

Note that the origin is an unstable equilibrium point of the first system in (4.66) and a stable equilibrium point of the second one. In the next section, we use a compliant contact model to investigate the system s motion in the regions of paradoxes. 

The linear eigenvalue analysis of Sect. 8.6 showed that when Co, Fq > 0 the origin is stable. More specifically, when the kinematic constraint instability conditions given by (8.21) are not satisfied and Co > 0, the trivial equilibrium point of the system is asymptotically stable. However, there can be situations where the region of attraction of the stable equilibrium point is quite small, leading to instabilities even when conditions of (8.21) do not hold. 

Since (8.22) has an exponentially stable equilibrium point at the origin, all solutions that start from initial conditions, satisfying r /( + tan l ku 0) + cii 0)) < 0 and m(0) > —Q and do not touch the 77 = 0 line, reach the origin (steady-sliding state) exponentially. If any of these trajectories reach the N = 0 line say at f = q, then the motion stops instantaneously and starts from the rest at u ti), —Q). This pattern continues and may result in a limit cycle at steady state. 

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