Stability analysis limited perturbations

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. 

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). 

To obtain a rough physical understanding of the mechanism of the instability, attention may be focused first on a planar detonation subjected to a one-dimensional, time-dependent perturbation. Since the instability depends on the wave structure, a model for the steady detonation structure is needed to proceed with a stability analysis. As the simplest structure model, assume that properties remain constant at their Neumann-spike values for an induction distance after which all of the heat of combustion is released instantaneously. If v is the gas velocity with respect to the shock at the Neumann condition, then may be expressed approximately in terms of the explosion time given by equation (B-57) as Z = vt. From normal-shock relations for an ideal gas with constant specific heats in the strong-shock limit, the Neumann-state conditions are expressible by v/vq = po/p —... 

With the surface of section technique, it can be observed that a limit cycle can undergo, for example, a Hopf bifurcation. When the stability analysis of the Poincare section is carried out, the real part of a complex conjugate pair of eigenvalues is seen to pass from negative to positive a small perturbation added to the limit cycle will evolve away from the cycle in an oscillatory fashion. This type of bifurcation results, then, in the appearance of a second... 

A Lyapunov exponent is a generalized measure trf the growth or decay of small perturbations away from a particular dynamical state. For perturbations around a fixed point or steady state, the Lyapunov exponents are identical to the stability eigenvalues of the Jacobian matrix discussed in an earlier section. For a limit cycle, the Lyapunov exponents are called Floquet exponents and are determined by carrying out a stability analysis in which perturbations are applied to the asymptotic, periodic state that characterizes the limit cycle. For chaotic states, at least one of the Lyapunov exponents will mm out to be positive. Algorithms for the calculation of Lyapunov exponents are discussed in a later section in conjunction with the analysis of experimental data. These algorithms can be used for simulations that yield possibly chaotic results as well as for the analysis of experimental data. 

The problem of existence and stability of limit cycles has been attadred in many ways. Luus and Lapidus have used an averaging technique, Douglas et employ a perturbation analysis, and Gilles and Hyon and Aris use Fourier transform methods to examine the occurrence of limit cycles, all with moderate successes. More recently Uppal et a/. have discussed the dynamic behaviour of the CSTR in terms of the bifurcation of the steady state, devdoping critmia for the existence and stability of oscillatory states as a function of system parameters, and... 

As was noted in Section 2.1.1, the concentration oscillations observed in the Lotka-Volterra model based on kinetic equations (2.1.28), (2.1.29) (or (2.2.59), (2.2.60)) are formally undamped. Perturbation of the model parameters, in particular constant k, leads to transitions between different orbits. However, the stability of solutions requires special analysis. Assume that in a given model relation between averages and fluctuations is very simple, e.g., (5NASNB) = f((NA), (A b)), where / is an arbitrary function. Therefore k in (2.2.67) is also a function of the mean values NA(t) and NB(t). Models of this kind are well developed in population dynamics in biophysics [70], Since non-linearity of kinetic equations is no longer quadratic, limitations of the Hanusse theorem [23] are lifted. Depending on the actual expression for / both stable and unstable stationary points could be obtained. Unstable stationary points are associated with such solutions as the limiting cycle in particular, solutions which are interpreted in biophysics as catastrophes (population death). Unlike phenomenological models treated in biophysics [70], in the Lotka-Volterra stochastic model the relation between fluctuations and mean values could be indeed calculated rather than postulated. 

We emphasize that the question of stability of a CA under small random perturbations is in itself an important unsolved problem in the theory of fluctuations [92-94] and the difficulties in solving it are similar to those mentioned above. Thus it is unclear at first glance how an analogy between these two unsolved problems could be of any help. However, as already noted above, the new method for statistical analysis of fluctuational trajectories [60,62,95,112] based on the prehistory probability distribution allows direct experimental insight into the almost deterministic dynamics of fluctuations in the limit of small noise intensity. Using this techique, it turns out to be possible to verify experimentally the existence of a unique solution, to identify the boundary condition on a CA, and to find an accurate approximation of the optimal control function. 

For both analyses, the procedure is to use dimensionless parameters in the set of differential equations describing the model, look for the steady state, investigate the linear stability, and determine the conditions for instability. Near the bifurcation values of the parameters, which initiate an oscillatory growing solution, a perturbation analysis provides an estimate for the period of the ensuing limit cycle behavior. 

In a film of infinite lateral extent, k can range from 0 to oo, so a necessary condition for instability is that AH > 2npgh. Since all wave numbers are available in a film of infinite extent, we see that this analysis predicts that the thin film will always be unstable, even with the stabilizing influence of surface tension, to disturbances of sufficiently large wavelength when van der Waals forces are present. Similarly, the Rayleigh Taylor instability that occurs when the film is on the underside of the solid surface will always appear in a film of infinite extent. In reality, of course, the thin film will always be bounded, as by the walls of a container or by the finite extent of the solid substrate. Hence the maximum wavelength of the perturbation of shape is limited to the lateral width, say W, of the film. This corresponds to a minimum possible wave number... 

One point that has not been emphasized is that all of the preceding analysis and discussion pertains only to the steady-state problem. From this type of analysis, we cannot deduce anything about the stability of the spherical (Hadamard Rybczynski) shape. In particular, if a drop or bubble is initially nonspherical or is perturbed to a nonspherical shape, we cannot ascertain whether the drop will evolve toward a steady, spherical shape. The answer to this question requires additional analysis that is not given here. The result of this analysis26 is that the spherical shape is stable to infinitesimal perturbations of shape for all finite capillary numbers but is unstable in the limit Ca = oo (y = 0). In the latter case, a drop that is initially elongated in the direction of motion is predicted to develop a tail. A drop that is initially flattened in the direction of motion, on the other hand, is predicted to develop an indentation at the rear. Further analysis is required to determine whether the magnitude of the shape perturbation is a factor in the stability of the spherical shape for arbitrary, finite Ca.21 Again, the details are not presented here. The result is that finite deformation can lead to instability even for finite Ca. Once unstable, the drop behavior for finite Ca is qualitatively similar to that predicted for infinitesimal perturbations of shape at Ca = oo that is, oblate drops form an indentation at the rear, and prolate drops form a tail. 

It is important to bear in mind that the predictions of Smb arise from a linear analysis, and so relate only to small perturbations about the equilibrium state. In conducting experiments to measure Smb it is therefore essential to take precautions to avoid equipment-induced disturbances that exceed the linear response limit of the system. Major disturbances can result from inefficient fluid distribution, so it is important to provide fluid stabilization before the distributor, and a sufficient pressure drop across it. Any bed internals that disrupt the flow path, such as thermometer pockets and heat-exchanger tubes, should be removed, and sources of mechanical vibration should be neutralized. 

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