Nonlinear stability analyses

Previous linear and nonlinear stability analyses for miscible displacement are reviewed. 

We now return to the base model (2,17)-(2.21) of frontal polymerization. We want to find uiuformly propagating FP waves and perform linear and nonlinear stability analyses, as we did in the case of the gasless combustion model. Before we study the model, we would like to reformulate it using the reaction front approximation. 

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. 

Unfortunately, most studies have not taken this approach. Numerical simulations of displacement have not been preceded by linear stability analyses to define the parameter limits within which an unstable displacement could be expected. [In their nonlinear stability analysis, Perrine and Gay (43) allowed their dispersion coefficients to be functions of velocity. In his linear stability analysis, Perrine (34, 35) assumed that his dispersion coefficients were constants.]... 

L. K. Gross and V. A. Volpert, Weakly nonlinear stability analysis of fronted polymerization, Smdies in Applied Mathematics, 110 (2003), pp. 351-375. 

Skeel, R., Srinivas, K. Nonlinear stability analysis of area-preserving integrators. SIAM J. Numer. Anal. 38,129-148 (2000). doi 10.1137/S0036142998349527... 

Wollkind, D.J. and Segel, L.A., Nonlinear stability analysis of the freezing of a dilute binary alloy, Philos. Trans. R. Soc. Lond. A268, 351, 1970. 

Figure4,7 The cross-shaped phase diagram for r 1/jit. Shown here are the four major regions and the critical values of A (solid lines) for the system of eqs. (4.2). Dashed curves are obtained from more detailed nonlinear stability analysis. (Adapted from Boissonade and De Kepper, 1980.)...

Dynamic nonlinear analysis techniques (Isidori 1995) are not directly applicable to DAE models but they should be transformed into nonlinear input-affine state-space model form by possibly substimting the algebraic equations into the differential ones. There are two possible approaches for nonlinear stability analysis Lyapunov s direct method (using an appropriate Lyapunov-function candidate) or local asymptotic stability analysis using the linearized system model. 

Shult and Volpert performed the linear stability analysis for the same model and confirmed this result [93]. Spade and Volpert studied linear stability for nonadiabatic systems [94]. Gross and Volpert performed a nonlinear stability analysis for the one-dimensional case [95]. Commissiong et al. extended the nonlinear analysis to two dimensions [96]. They confirmed that, unlike in SHS [97], uniform pulsations are difficult to observe in FP. In fact no such one-dimensional pulsating modes have been observed. 

Gross, L.K. and Volpert, VA. (2003) Weakly nonlinear stability analysis of frontal polymerization. Stud. Appl. Math., 110, 351-376. 

These experimentally detected combustion modes were analytically predicted follo-v fing a nonlinear stability analysis of the set of equations governing the combustion process (essentially the energy conservation in the condensed phase with appropriate initial and boundary conditions). This nonlinear analysis accounts for the influence of the properties of the burning material and the ambient conditions (included pressure and diabaticity), allowing to predict PDL and the values of pressure and radiant flux intensity originating oscillatory combustion. Moreover, several numerical checks of the analytical predictions were performed by numerical integration of the basic set of equations under the appropriate ambient conditions. Both the numerical checks and experimental results fully confirm the validity of the analytical predictions. 

The nonlinear modes are usually referred to as solutions to the nonlinear Helmholtz equation in the waveguide cross-section. For their investigation, power-dispersion diagrams are commonly used that give values of critical powers and are helpful in stability analysis of the fundamental mode. ... 

Of the various methods of weighted residuals, the collocation method and, in particular, the orthogonal collocation technique have proved to be quite effective in the solution of complex, nonlinear problems of the type typically encountered in chemical reactors. The basic procedure was used by Stewart and Villadsen (1969) for the prediction of multiple steady states in catalyst particles, by Ferguson and Finlayson (1970) for the study of the transient heat and mass transfer in a catalyst pellet, and by McGowin and Perlmutter (1971) for local stability analysis of a nonadiabatic tubular reactor with axial mixing. Finlayson (1971, 1972, 1974) showed the importance of the orthogonal collocation technique for packed bed reactors. 

P 21.4 Stability Analysis of Steady-States of Nonlinear Systems Analyze the stability of the steady-state C, = 0 of Figs. 21.6a and b. 

In many cases ordinary differential equations (ODEs) provide adequate models of chemical reactors. When partial differential equations become necessary, their discretization will again lead to large systems of ODEs. Numerical methods for the location, continuation and stability analysis of periodic and quasi-periodic trajectories of systems of coupled nonlinear ODEs (both autonomous and nonautonomous) are extensively used in this work. We are not concerned with the numerical description of deterministic chaotic trajectories where they occur, we have merely inferred them from bifurcation sequences known to lead to deterministic chaos. Extensive literature, as well as a wide choice of algorithms, is available for the numerical analysis of periodic trajectories (Keller, 1976,1977 Curry, 1979 Doedel, 1981 Seydel, 1981 Schwartz, 1983 Kubicek and Hlavacek, 1983 Aluko and Chang, 1984). 

XPPAUT (http //www.math.pitt.edu/ bard/xpp/xpp.html) offers deterministic simulations with a set of very good stiff solvers. It also offers fitting, stability analysis, nonlinear systems analysis, and time-series analysis, like histograms. The GUI is simple. It is mainly available under Linux, but also mns on Windows. 

Many methods have been developed for model analysis for instance, bifurcation and stability analysis [88, 89], parameter sensitivity analysis [90], metabolic control analysis [16, 17, 91] and biochemical systems analysis [18]. One highly important method for model analysis and especially for large models, such as many silicon cell models, is model reduction. Model reduction has a long history in the analysis of biochemical reaction networks and in the analysis of nonlinear dynamics (slow and fast manifolds) [92-104]. In all cases, the aim of model reduction is to derive a simplified model from a larger ancestral model that satisfies a number of criteria. In the following sections we describe a relatively new form of model reduction for biochemical reaction networks, such as metabolic, signaling, or genetic networks. 

Such nested applications of single-parameter singular perturbation theory (i.e., the extension of the analysis of two-time-scale systems presented in Chapter 2 to multiple-time-scale systems) have been used for stability analysis of linear (Ladde and Siljak 1983) and nonlinear (Desoer and Shahruz 1986) systems in the standard form. However, as emphasized above (Section 2.3), the ODE models of chemical processes are most often in the nonstandard singularly perturbed form, with the general multiple-perturbation representation... 

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. 

Sastry Nonlinear Systems Analysis, Stability, and Control... 

Linear stability analysis does not provide information on how a system will evolve when a state becomes unstable. It does not distinguish between metastable and stable states when multiple local states are possible for given boundary conditions. Boundary conditions affect the value of the Lyapunov functional, and cause changes between stable and metastable states, hence altering the relative stability. An unstable state corresponds to the saddle points of the functional and defines a barrier between the attractors. Approximate solutions of nonlinear evolution equations may help us to understand how the system will behave in time and space. 

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. 

Instead of using this equation, in the literature, there are few models proposed by which the frequency or Strouhal number of the shedding is fixed. Koch (1985) proposed a resonance model that fixes it for a particular location in the wake by a local linear stability analysis. Upstream of this location, flow is absolutely unstable and downstream, the flow displays convective instability. Nishioka Sato (1973) proposed that the frequency selection is based on maximum spatial growth rate in the wake. The vortex shedding phenomenon starts via a linear instability and the limit cycle-like oscillations result from nonlinear super critical stability of the flow, describ-able by Eqn. (5.3.1). 

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.]... 

For the reader who is knowledgeable about the dynamic s of nonlinear systems, it will not be surprising that the stability analysis predicts an exchange of stability for steady solutions of (4-213) as we pass through the limit point from the lower to the upper solution branch. This behavior is expected on general grounds from the theories of nonlinear dynamical systems (see Iooss and Joseph).21... 

Nonlinear equations, discrete algebraic equations, continuation and stability analysis, parameter analysis, and parameter estimation... 

Lakshmikantham, V, Leela, S., and Martynyuk, A. A., Stability Analysis of Nonlinear Systems, New York Marcel Dekker, 1989. 

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