Stability analysis

Stability Analysis. - In the last decade there has been great interest in the numerical solution of special second order periodic initial-value problems (see refs. 6, 7 and references therein) 

In order to investigate the periodic stability properties of numerical methods for solving the initial-value problem (1) Lambert and Watson introduced the scalar test equation 

Based on the theory developed in ref 8, when a linear symmetric multistep method 

The general solution of the above difference equation is given by  

We note here that the roots of the polynomial (65) are perturbations of the roots of p. We denote as and Q2 the perturbations of the principal roots of p. 

In this section, we discuss the stability of finite difference approximations using the well-known von Neumann procedure. This method introduces an initial error represented by a finite Fourier series and examines how this error propagates during the solution. The von Neumann method applies to initial-value problems for this reason it is used to analyze the stability of the explicit method for parabolic equations developed in Sec. 6.4.2 and the explicit method for hyperbolic equations developed in Sec. 6.4.3. 

Numerical Solution of Partial Differential Equations Chapter 6 

Define the error as the difference between the solution of the finite difference approximation and the exact solution of the differential equation at step (m, n)  

The explicit finite difference solution (6.58) of the parabolic partial differential equation (6.18) 

This is a nonhomogeneous finite difference equation in two dimensions, representing the propagation of error during the numerical solution of the parabolic partial differential equation (6.18). The solution of this finite difference equation is rather difficult to obtain. For this reason, the von Neumann analysis considers the homogeneous part of Eq. (6.126)  

Every dynamic adaptronic system must be checked for stability in the case of disturbances. For linear elastic adaptronic structures, asymptotic stability as defined in Sect. 5.2.4 is guaranteed if the poles (or eigenvalues) of the closed-loop active system lie in the left complex half-plane, i.e. if they have negative real parts. More stringent stability criteria, such as the generalized Nyquist criterion [7], also consider the zeros of the adaptronic system. 

In the case of nonlinear systems that cannot be reduced to a linearized system, stability is much more difficult to assess. Lyapunov s direct method [1] requires a suitable energy function to be found. Often, only numerical time integration gives an indication of the dynamic behaviour and stability that cannot be proven otherwise. 

The following two definitions are necessary to clarify the subsequent study of the linear stability in dynamical systems. 

Definition 3.1. (Lyapunov Stability) Consider the autonomous system (3.2) in a neighborhood D c / of y = 0. The equilibrium point y = 0 is called stable (in the sense of Lyapunov) if for each e 0 there exists 6(b) 0 such that y(0) 6 yields y(f) e for all f 0. The equilibrium point is unstable, otherwise.  

Let (t) = ae be the solution of (3.8) where a is a constant n-vector and rj is a. parameter to be determined. Substituting this solution into (3.8) and simplifying 

For (3.25) to have nontrivial solutions (i.e., a 0), the matrix rfM + rjAy + Ap must be singular. Thus, the characteristic equation is obtained as 

Assuming the general complex-valued eigenvalue as pj = pj + ieoj, where pj and ojj are real numbers, the following two types of instability are identified [48]  

The prediction of a threshold current above which a leveler is unable to stabilize growth has also been identified in a recent linear stability study that includes two additional terms not previously considered, namely (a) complex formation between the additive and the metal cation and (b) interaction between the polar additives and 

Nonequilibrium conditions may occur with respect to disturbances in the interior of a system, or between a system and its surroundings. As a result, the local stress, strain, temperature, concentration, and energy density may vary at each instance in time. This may lead to instability in space and time. Constantly changing properties cannot be described properly by referring to the system as a whole. Some averaging of the properties in space and time is necessary. Such averages need to be clearly stated in the utilization and correlation of experimental data, especially when their interpretations are associated with theories that are valid at equilibrium. Components of the generalized flows and the thermodynamic forces can be used to define the trajectories of the behavior of systems in time. A trajectory specifies the curve represented by the flow and force components as functions of time in the flow-force space. 

The equations of motion (5.53, 54) can now be linearized with respect to (t) nd /(t). This leads to the following, which are valid only in the vicinity of P(d,x)  

The singular point is a stable focus if the eigenvalues Aj and A2 both have negative real parts since then both (r) and tj (t) approach zero with increase in time. On the other hand, the singular point is unstable if the real part of at least one eigenvalue is positive.  

It is reasonable - and confirmed by the calculations presented in Sect. 5.5 - to assume relative large values for the trend reversal parameter / and the strategic choice amplitude Thus 

From (5.74) it follows immediately that there exists a stable focus at the origin of the d-x plane under the condition 

Similarly to the spherical micelle problem, at the CSAC the concentration of hemi-micelles, (]) , is a minimum. From Equation 5.3, Qm must be positive, and also 0f2m/ 

From the SC F calculations it is further possible to obtain the surface tension y. The difference from that of the pristine interface determines the surface pressure. We have evaluated the surface pressure at the CSAC, n, and the corresponding area per adsorbed surfactant molecule a. These quantities are also included in Table 5.3. 

The inhomogeneities at the interface appear at lower surface pressure and higher area per molecule upon increasing the ionic strength (Table 5.3). 

As the counterions penetrate the head group region to directly compensate the charge of the surfactant head, it is natural to expect that the size of the ion has a large 

---

A linear stability analysis of (A3.3.57) can provide some insight into the structure of solutions to model B. The linear approximation to (A3.3.57) can be easily solved by taking a spatial Fourier transfomi. The result for the Ml Fourier mode is... 

Wisdom, J., Holman, M. Symplectic Maps for the n-Body Problem Stability Analysis. Astron. J. 104 (1992) 2022-2029... 

To pursue this question we shall examine the stability of certain steady state solutions of Che above equaclons by the well known technique of linearized stability analysis, which gives a necessary (but noc sufficient) condition for the stability of Che steady state. 

The described method can generate a first-order backward or a first-order forward difference scheme depending whether 0 = 0 or 0 = 1 is used. For 9 = 0.5, the method yields a second order accurate central difference scheme, however, other considerations such as the stability of numerical calculations should be taken into account. Stability analysis for this class of time stepping methods can only be carried out for simple cases where the coefficient matrix in Equation (2.106) is symmetric and positive-definite (i.e. self-adjoint problems Zienkiewicz and Taylor, 1994). Obviously, this will not be the case in most types of engineering flow problems. In practice, therefore, selection of appropriate values of 6 and time increment At is usually based on trial and error. Factors such as the nature of non-linearity of physical parameters and the type of elements used in the spatial discretization usually influence the selection of the values of 0 and At in a problem. 

Linear stability analysis has been successfully applied to derive the critical Marangoni number for several situations. 

Bai [48] presents a linear stability analysis of plastic shear deformation. This involves the relationship between competing effects of work hardening, thermal softening, and thermal conduction. If the flow stress is given by Tq, and work hardening and thermal softening in the initial state are represented... 

Figure B-i5 shows some of the terms encountered in stability analysis.

Venable, H. Dean, The K Factor A New Mathematical Tool for Stability Analysis and Synthesis, POWERCON March 1983. 

The field of ehemieal kineties and reaetion engineering has grown over the years. New experimental teehniques have been developed to follow the progress of ehemieal reaetions and these have aided study of the fundamentals and meehanisms of ehemieal reaetions. The availability of personal eomputers has enhaneed the simulation of eomplex ehemieal reaetions and reaetor stability analysis. These aetivities have resulted in improved designs of industrial reaetors. An inereased number of industrial patents now relate to new eatalysts and eatalytie proeesses, synthetie polymers, and novel reaetor designs. Lin [1] has given a eomprehensive review of ehemieal reaetions involving kineties and meehanisms. 

K. Sekimoto, R. Oguma, K. Kawasaki. Morphological stability analysis of partial wetting. Ann Phys 776 359-392, 1987. 

A. Valance. Porous silicon formation Stability analysis of the silicon-electrolyte interface. Phys Rev B 52 8323, 1995. 

The predicted energy, which appears in the SCF summary section preceding the stability analysis output, is -149.61266 hartrees, which is about 53.5 kcal/mol lower than that corresponding to the RHF wavefunction (-149.52735). 

For the phase stability analysis we follow the method given by Kanamori and Kakehashi of geometrical inequalities and compute the antiphase boundary energy defined by... 

At the critical value a = oi = 1, however, becomes unstable and the a-dependent fixed point becomes stable. This exchange of stability between two fixed points of a map is known as a transcritical bifurcation. By using the same linear-stability analysis as above, we see that remains stable if — 1 < a(l — Xjjj) < 1, or for all a such that 1 < a < 3. Something more interesting happens at a — 3. 

Develop final Perform structural design of analysis of component acceptable accuracy Determine structural response—stresses, support reactions, deflections, and stability—based on a structural analysis of acceptable accuracy. Determine acceptable accuracy based on economic value of component, consequences of failure, state-of-the-art capability in stress and stability analysis, margin of safety, knowledge about loads and materials properties, conservatism of loads, provisions for further evaluation by prototype testing... 

The conditions which lead a homogeneous fluid mixture to split into two separate fluid phases can be described by classical thermodynamic stability analysis as discussed in numerous textbooks.9 Such analysis has often been... 

Izumisawa, S. and Jhon, M. S., "Stability Analysis and Molecular Simulation of Nanoscale Lubricant Films with Chain-End Functional Groups, /. Appl. Phys., 2002, Vol. 91,2002, pp. 7583-7585. 

Borgaonkar H. and Ramani K., Stability analysis in single screw extrusion of thermoplastic elastomers using simple design of experiments, Adv. Polym. Technol., 17, 115, 1998. 

Stability analysis. We now investigate the stability of scheme (4) with respect to initial data in the case of homogeneous boundary conditions and zero right-hand side of the equation. A reasonable statement of the problem is... 

When the transition operator happens to be constant, the stability analysis with respect to the initial data is mostly based on estimates of the norms of the transition operator. 

The problem statement. We pursue the stability analysis of two-layer schemes by having recourse to their canonical form... 

Stability and convergence. The general stability theory for two-layer schemes applies equally well to the stability analysis of the weighted scheme (7). With this aim, the appropriate difference scheme with the homogeneous boundary conditions comes first ... 

This means that the second term O(r ) in the available exppression for p. will be excluded from further consideration. The proof of this statement is omitted here. As a matter of fact, the stability of the scheme at hand with respect to the boundary conditions is revealed through such a stability analysis. 

Further stability analysis is connected with the homogeneous bouncl-... 

Stability analysis is mostly based on the assumptions that the bound-... 

The question of the accuracy of the scheme, being of principal importance in the theory, amounts to studying the error of approximation and stability of the scheme. Stability analysis neces.sitates imposing a priori estimates for the difference problem solution in light of available input data. This is a problem in itself and needs investigation. 

Stability theory is the central part of the theory of difference schemes. Recent years have seen a great number of papers dedicated to investigating stability of such schemes. Many works are based on applications of spectral methods and include ineffective results given certain restrictions on the structure of difference operators. For schemes with non-self-adjoint operators the spectral theory guides only the choice of necessary stability conditions, but sufficient conditions and a priori estimates are of no less importance. An energy approach connected with the above definitions of the scheme permits one to carry out an exhaustive stability analysis for operators in a prescribed Hilbert space Hh-... 

X. Chen, C. P. Tan, and C. M. Haberfield. Wellbore stability analysis guidelines for practical well design. In Proceedings Volume, pages 117-126. SPE Asia Pacific Oil Gas Conf (Adelaide, Australia, 10/28-10/31), 1996. 

AUTOREFRIGERATED REACTOR OF LUYBEN DYNAMIC STABILITY ANALYSIS... 

Bauernschmitt, R., Ahlrichs, R., 1996a, Stability Analysis for Solutions of the Closed Shell Kohn-Sham Equation , J. Chem. Phys., 104, 9047. 

---