Stability eigenvalue analysis

The stability characteristics of the steady-state points is then determined via an eigenvalue analysis of the linearized version of the two DE (7.198) and (7.199). The linearized form of equations (7.198) and (7.199) is as follows ... 

Stability is determined by the eigenvalue analysis at an equilibrium point for flows and by the characteristic multiplier analysis of a periodic solution at a fixed point for maps [3]. 

Certain quantitative measures from linear control theory may help at various steps to assess relationships between the controlled and manipulated variables. These include steady-state process gains, open-loop time constants, singular value decomposition, condition numbers, eigenvalue analysis for stability, etc. These techniques are described in... 

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. 

An eigenvalue analysis is performed after every 5 increments to determine the residual stability of the system. Buckling can be detected as the moment when the first vibration frequency of the system becomes zero. 

Although the linear complex eigenvalue analysis method is useful in establishing the local stability boundaries of the equilibrium point in the system s parameter space, it does not reveal any information regarding dynamic behavior of the system. Further investigations such as numerical simulations or nonlinear analysis methods may be utilized to study the amplitude and frequency of the resulting vibrations under the mode coupling instability mechanism. 

Expanding the sample size to 2Xc admits the other shape families shown on Fig. 6 into the analysis and leads to additional codimension-two interactions between the shapes is the (1A<.)- family and shapes with other numbers of cells in the sample. The bifurcation diagram computed for this sample size with System I and k = 0.865 is shown as Fig. 11. The (lAc)- and (Ac/2)-families are exactly as computed in the smaller sample size, but the stability of the cell shapes is altered by perturbations that are admissible is the larger sample. The secondary bifurcation between the (lAc)- and (2Ae/3)-families is also a result of a codimension two interaction of these families at a slightly different wavelength. Two other secondary bifurcation points are located along the (lAc)-family and may be intersections with the (4Ac and (4A<./7) families, as is expected because of the nearly multiple eigenvalues for these families. 

The basic idea is very simple In many scenarios the construction of an explicit kinetic model of a metabolic pathway is not necessary. For example, as detailed in Section IX, to determine under which conditions a steady state loses its stability, only a local linear approximation of the system at this respective state is needed, that is, we only need to know the eigenvalues of the associated Jacobian matrix. Similar, a large number of other dynamic properties, including control coefficients or time-scale analysis, are accessible solely based on a local linear description of the system. 

A standard stability analysis of the equilibrium E(u0,p0) implies that the latter is linearly stable for i < ic and unstable for > ic. Indeed, for fixed the eigenvalues (T 2 of the appropriate linearized problem are given by the expression... 

We have now seen how local stability analysis can give us useful information about any given state in terms of the experimental conditions (i.e. in terms of the parameters p and ku for the present isothermal autocatalytic model). The methods are powerful and for low-dimensional systems their application is not difficult. In particular we can recognize the range of conditions over which damped oscillatory behaviour or even sustained oscillations might be observed. The Hopf bifurcation condition, in terms of the eigenvalues k2 and k2, enabled us to locate the onset or death of oscillatory behaviour. Some comments have been made about the stability and growth of the oscillations, but the details of this part of the analysis will have to wait until the next chapter. 

The local stability and character of the singular points can be determined by the usual analysis of the eigenvalues of the Jacobian matrix... 

The simplest new phenomenon induced by this mechanism is a secondary bifurcation from the first primary branch, arising from the interaction between the latter and another nearby primary branch. It leads to the loss of stability of the first primary branch or to the stabilization of one of the subsequent primary branches, as illustrated in Fig. 1. The analysis of this branching follows similar lines as in Section I. A, except that one has now two control parameters X and p., which are both expanded [as in equation (5)] about the degeneracy point (X, p.) corresponding to a double eigenvalue of the linearized operator L. Because of this double degeneracy, the first equation (7) is replaced by... 

For the low-temperature steady state y in Figure 4 (A-2) a similar analysis shows that this steady state is stable as well. However, for the intermediate steady-state temperature yi and Sy > 0 the heat generation is larger than the heat removal and therefore the system will heat up and move away from y2. On the other hand, if 5y < 0 then the heat removal exceeds the heat generation and thus the system will cool down away from 2/2 -We conclude that yi is an unstable steady state. For 2/2, computing the eigenvalues of the linearized dynamic model is not necessary since any violation of a necessary condition for stability is sufficient for instability. 

The condition for stability of the ground state is that E[p] be a convex functional and therefore that rf r, r ), twice its Hessian, be positive-definite. r/ (r, r ) thus possesses a minimum eigenvalue and s (r, r ) a maximum, sj. A perturbation theoretic analysis sketched briefly in [3] then establishes that the upper bound of the spectrum of t]x(r, r ) is oo, and therefore the lower bound of the s is zero. 

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. 

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. 

Experiments on transition for 2D attached boundary layer have revealed that the onset process is dominated by TS wave creation and its evolution, when the free stream turbulence level is low. Generally speaking, the estimated quantities like frequency of most dominant disturbances, eigenvalues and eigenvectors matched quite well with experiments. It is also noted from experiments that the later stages of transition process is dominated by nonlinear events. However, this phase spans a very small streamwise stretch and therefore one can observe that the linear stability analysis more or less determines the extent of transitional flow. This is the reason for the success of all linear stability based transition prediction methods. However, it must be emphasized that nonlinear, nonparallel and multi-modal interaction processes are equally important in some cases. 

The Bromwich contour for point A was chosen in the a -plane on a line extending from -20 to 4-20 that is below and parallel to the areal axis at a distance of 0.009 and in the w-plane it extended from -1 to - -1, above and parallel to the u>reai axis at a distance of 0.02. For the other points, the Bromwich contour in the a- plane is located at a distance of 0.001 below the Ureal axis. The choice of the Bromwich contour in the a- plane was such that all the downstream propagating eigenvalues lie above it. Orr-Sommerfeld equation was solved along these contours with 8192 equidistant points in the a- plane and 512 points in the w-plane. Orr-Sommerfeld equation was solved taking equidistant 2400 points across the shear layer in the range 0 < 2/ < 6.97. Spatial stability analysis produced waves for the four points of Fig. 4.2 with the properties shown in Table 4.1. 

This completes the definition of the stability problem for the mixed convection flow over the horizontal plate. For a given K and Re, one would be required to solve (6.4.19)-(6.4.38), starting with the initial conditions (6.4.39)-(6.4.58) and satisfy (6.4.63) for particular combinations of the eigenvalues obtained as the complex k and u>. We will use the procedure adopted in Sengupta et al. (1994) to obtain the eigen-spectrum for the mixed convection case, when the problem is in spatial analysis framework. In the process, it is possible to scan for all the eigenvalues in a limited part of the complex k- plane, without any problem of spurious eigenvalues. 

The analysis of the stability of isolated stationary points is different in the phase-space treatment from that in the coordinate space treatment. In the coordinate space treatment the slope of the potential energy surface gives the forces exerted on the system. Stationary points occur at extrema of the potential energy. Their stability is determined by the eigenvalues of the matrix of second derivatives evaluated at the extremum. Assuming the system has n DOFs, it will possess n... 

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