Linear stability analysis perturbations

Aiming at a more formal analysis, the asymptotic stability of a steady-state value S° of a metabolic system upon an infinitesimal perturbation is determined by linear stability analysis. Given a metabolic system at a positive steady-state value... 

Here C is defined by the boundary value in the case of the Dirichlet conditions (3.1.3b), (3.1.3d) at one of the end points or by the space averages of the initial concentrations in the case of the Neumann conditions (3.1.3a), (3.1.3c) at both ends. In the spirit of a standard linear stability analysis consider a small perturbation of the equilibrium of the form... 

We write the solution as the vector X = (6,(j),u,vx,vy,i ,P,) consisting of the angular variables of the director, the layer displacement, the velocity field, the pressure, and the modulus of the (nematic or smectic) order parameter. For a spatially homogeneous situation the equations simplify significantly and the desired solution Xo can directly be found (see Sect. 3.1). To determine the region of stability of Xq we perform a linear stability analysis, i.e., we add a small perturbation Xi to... 

Because the model (11.8) that describes this physiological process is nonlinear, we cannot answer these questions in total generality. Rather, we must be content with understanding what happens when we make a small perturbation on the states x, y, and 2 away from the equilibrium. The fact that we are assuming that the perturbation is small allows us to carry out what is known as linear stability analysis of the equilibrium state. 

How the perturbations affect the state of the system depends on the eigenvalues Kk. If any eigenvalue has a positive real part, then the solution x grows exponentially, and the corresponding eigenvectors are known as unstable modes. If, on the other hand, all the eigenvalues have negative real parts then a perturbation around the stationary state exponentially decays and the system returns back to its stable state. The linear stability analysis is valid for small perturbations ( x / A v 1) only. 

The front is inherently unstable, however, and this is often studied by a linear stability analysis. Infinitesimal perturbations are applied to all of the variables to simulate reservoir heterogeneities, density fluctuations, and other effects. Just as in the Buckley-Leverett solution, the perturbed variables are governed by force and mass balance equations, and they can be solved for a perturbation of any given wave number. These solutions show whether the perturbation dies out or if it grows with time. Any parameter for which the perturbation grows indicates an instability. For water flooding, the rate of growth, B, obeys the proportionality... 

In order to understand this instability problem, the first step is to construct a linear stability analysis. This is used to define the parameter limits within which instabilities can be triggered by infinitesimal perturbations to the system. Within these parameter limits, a nonlinear stability analysis should be used to study the development of these instabilities. In this way, one can determine the parameter limits within which instabilities may be of practical concern. 

Christie and Bond ( 4) began with a linear stability analysis, but they did not construct a stability curve to define the parameter domain in which instabilities could be expected. In their nonlinear analysis, instabilities were initiated by random perturbations in the initial concentration distribution at the entrance (macroscopic perturbations). [Peters and Kasap (45) used the same method to initiate instabilities.]... 

First, since most of these studies were not preceded by a linear stability analysis to define the parameter limits within which an unstable displacement could be expected, we are free to speculate that in some cases the displacements were actually stable to infinitesimally small perturbations (but not necessarily stable to the macroscopic perturbations). 

According to the theory of linear stability analysis, infinitesimally small perturbations are superimposed on the variables in the steady state and their transient behavior is studied. At this stage the difference between turbulent fluctuations and perturbations may be noted. Turbulence is the characteristic feature of the multiphase flow under consideration the mean and fluctuating quantities were given by Eq. (2). The fluctuating components result in eddy diffusivity of momentum, mass, and Reynolds stresses. The turbulent fluctuations do not alter the mean value. In contrast, the perturbations are superimposed on steady-state average values and another steady... 

The following calculations are based on a linear stability analysis taking into account only the first order of. Furthermore, we use two approximations. One is the long wavelength approximation the wavelength of perturbation of the solid-liquid interface is much larger than the mean thickness of the liquid film, then we can define a small dimensionless wavenumber The other is the quasistationary approximation we... 

We assume that the flat interface between the two fluids, now designated as z = 0, is perturbed with an arbitrary infinitesimal perturbation of shape. As usual, for a linear stability analysis, we consider only a single Fourier mode in each of the x and y directions, with the wave number (or wavelength) as a parameter in the stability analysis. Hence we consider a perturbation of the form... 

Note that in fact the plane Poiseuille flow (1.17) is also an exact solution of the full Navier-Stokes equation. However, it was shown by linear stability analysis that this becomes unstable to small perturbations at a critical Reynolds number of 5772. In fact, the transition to turbulence is observed experimentally at even lower values of Re around 1000. 

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 linear stability analysis of the stationary distorted state above the OFT induced by obliquely incident, linearly polarized light was repeated in [12], this time without the numerous approximations used in earlier works, using numerical methods. In addition, dye-doped nematics were considered, for which the threshold for the OFT is much lower and thus permits the experimental realization of a light beam that is much wider than the cell width. For this case, however, absorption also has to be taken into account, which means that the OFT does not happen at p = 1 for perpendicular incidence (see Fig. 16). Taking all of this into account, the exact reorientation profiles for the director were calculated numerically first, and its stability was investigated with respect to spatially periodic perturbations (proportional to exp[i qx + py)]). Then the... 

A linear stability analysis [21, 24—26] leads to the establishment of the dispersion relation for the system relating the time constant t with the wave vector of a sinusoidal perturbation of the film ... 

In searching for the necessary conditions under which the smooth-stratified flow configuration is stable, linear stability analysis is carried out on the transient two-fluid continuity and momentum Equations 1, 2, 7. The equations are perturbed around the smooth fully developed stratified flow pattern. Following the route of temporal stability analysis h = heKkx-m). gi(icx-[Pg.327]

When we studied the emergence of temporal oscillations in Chapter 2, we found that it was useful to examine whether a small perturbation to a steady state would grow or decay. We now attempt a similar linear stability analysis of a system in which diffusion, as well as reaction, can occur. First, consider the general reaction-diffusion equation ... 

As in the homogeneous case, linear stability analysis can tell us whether a small perturbation will grow, but it does not reveal what sort of state the system will evolve into. In Figure 6,1, we illustrate the range of m values over which instability occurs in the Brusselator. 

In Chapter 2, we introduced linear stability analysis as a way of gaining important information about the dynamical behavior of a system of DDEs by studying how the system responds to small perturbations of its steady state(s). An analogous approach can be applied to systems of DDEs. The analysis is similar, but significantly more difficult. 

The linear stability analysis developed in Chapter 2 implies that the steady state will be stable to spatially uniform perturbations if, and only if. 

We would like to know if this stationary solution will be stable to small perturbations Xj. Linear stability analysis provides the answer in the following way. Consider a small perturbation Xk . 

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