Bifurcations and Stability Analysis

Consider, first, only steady state solutions of a dynamical system. As in the examples in the previous section, the dynamical system may be a single ordinary differential equation (ODE) such as a rate equation or population growth equation, in the case where there is a single dynamical variable. If there is more than one dynamic variable, the dynamical system consists of the same number of usually coupled ODEs. In the former case of a single variable, the dynamical system is thus given by a single equation of die form 

We let X be a steady state solution of the dynamical equation, that is, x satisfies  

As we have seen, there may be more than one solution to Eq. [12]. We will consider each steady state solution in turn, that is, we will carry out a local stability analysis for each that satisfies Eq. [12]. Because the local analysis will involve a linearization of the full equations, this type of analysis is also called a linear stability analysis. 

The stability analysis begins by applying a small perturbation, 11, to the steady state solution of interest  

The variable x in Eq. [13] is a function of time because the perturbation, ti, is a function of time. The time dependence of the variable x is given, of course, by Eq. [11], because the general dynamical equation of motion applies to any x. 

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Micromixing effects in the Nicolis-Puhl reaction Numerical bifurcation and stability analysis of the IEM model. Chemical Engineering Science 46, 1829-1847. 

Fox, R. O., G. Erjaee, and Q. Zou (1994). Bifurcation and stability analysis of micromixing effects in the chlorite-iodide reaction. Chemical Engineering Science 49, 3465-3484. 

R. Seydel. Practical Bifurcation and Stability Analysis From Equilibrium, to Chaos. Springer-Verlag, New York, 2nd. edition, 1994. 

The interest in periodically forced systems extends beyond performance considerations for a single reactor. Stability of structures and control characteristics of chemical plants are determined by their responses to oscillating loads. Epidemics and harvests are governed by the cycle of seasons. Bifurcation and stability analysis of periodically forced systems is especially important in the... 

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. 

Seydel Practical Bifurcation and Stability Analysis From Equilibrium to Chaos... 

Computational techniques are centrally important at every stage of investigation of nonlinear dynamical systems. We have reviewed the main theoretical and computational tools used in studying these problems among these are bifurcation and stability analysis, numerical techniques for the solution of ordinary differential equations and partial differential equations, continuation methods, coupled lattice and cellular automata methods for the simulation of spatiotemporal phenomena, geometric representations of phase space attractors, and the numerical analysis of experimental data through the reconstruction of phase portraits, including the calculation of correlation dimensions and Lyapunov exponents from the data. 

Mohl et al. [21, 22] implemented a dynamic EQ model (with Murphree type efficiencies) in the DIVA simulator and carried out a numerical bifurcation and stability analysis on the MTBE and TAME processes. They also show that the window of opportunity for MSS to actually occur in the MTBE process is quite small. 

Seydel, R. 1994. Practical Bifurcation and Stability Analysis., 2nd ed. Springer-Verlag New York. 

Burrougbs, E., 2003. Convection in a Thermosypbon, Bifurcation and Stability Analysis (Pb.D. dissertation). Tbe University of New Mexico, Albuquerque, New Mexico. 

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). 

Methods are being explored for enabling microscopic simulators to perform system-level analysis—mainly numerical bifurcation and stability analy-... 

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. 

There are two more important advantages of these models. One is that it is possible under some conditions to carry out an exact stability analysis of the nonequilibrium steady-state solutions and to determine points of exchange of stability corresponding to secondary bifurcations on these branches. The other is that branches of solutions can be calculated that are not accessible by the usual approximate methods. We have already seen a case here in which the values of parameters correspond to domain 2. This also happens when the fixed boundary conditions imposed on the system are arbitrary and do not correspond to some homogeneous steady-state value of X and Y. In that case Fig. 20 may, for example,... 

Equations (3.20) and (3.21) with their stationary-state solutions (3.24) and (3.25) are simple enough to provide a good introduction to some of the mathematical techniques which can serve us so well in analysing these sorts of chemical models. In the next sections we will explain the ideas of local stability analysis ( 3.2) and then apply them to our specific model ( 3.3). After that we introduce the basic aspects of a technique known as the Hopf bifurcation analysis ( 3.4) which enables us to locate the conditions under which oscillatory states are likely to appear. We set out only those aspects that are required within this book, without any pretence at a complete... 

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. 

This argument shows that for the first-order reaction model the stationary state always has some sort of stability to perturbations. In fact, this is only a first step and will not reveal Hopf bifurcations or oscillatory solutions, should they occur-. A full stability analysis of typical flow-reaction schemes will appear in the next chapter. 

Local stability analysis and Hopf bifurcation for n variables... 

The dynamic behavior of this system was examined using linear stability analysis [8] and is thoroughly discussed in the review articles [9, 10], The results can be best summarized in a so-called two-parameter bifurcation diagram, in which, similar to phase diagrams, regions with qualitatively different behavior (states) are indicated. The dominant regimes of the N-NDR oscillator (Eqs. (38a,b)) are depicted in Fig. 11... 

This argumentation can be easily extended to two-variable (H)N-NDR systems. Performing a linear stability analysis of a homogeneous stationary state, it is straightforward to show that a homogeneous stationary state can never become unstable in a nontrivial Hopf bifurcation with n = 1. Thus, whenever a Hopf bifurcation occurs, a homogeneous limit cycle is born. Standing waves and pulses are therefore not to be expected under current control. 

Mallubhotla, H. Edelstein, W.A. Earley, T.A. Belfort, G. Magnetic resonance flow imaging and numerical analysis of curved tube flow. 16. Effect of curvature and flow rate on Dean vortex stability and bifurcation. AIChE J. 2001, 47, 1126-1140. 

The functions f c, z,p) and g c) include the transport fluxes of cytoplasmic Ca2+ across the ER and plasma membrane, w c,p) the production and degradation of IP3. For realistic functions /, g and w the existence of a limit cycle must generally be shown numerically. However, the local stability properties of the steady state give an idea. One can show that, due to the Ca " transport across the plasma membrane, there is a unique steady state (c, z, p) [23]. Therefore, changes in stability of the unique steady state are likely to be connected with an Hopf bifurcation and the birth/death of a limit cycle. Generally, if the steady state is unstable it is to be expected that the trajectories move toward a stable limit cycle. Note that a stable limit cycle and a stable steady state can coexist. Our analysis can make no predictions in this regard. 

We perform a stability analysis of the subspace Mg. First, we present the two-dimensional bifurcation diagram for the stability of CW solutions in Fig. 6.15(a). One can identify different bifurcations Hopf, Fold, and... 

In 1974, McNeil and Walls [11] applied to the logistic model the same techniques of stability analysis that Prigogine and his group had used for the case of the Brusselator. As shown in Appendix B, they proved the existence of a bifurcation at K = 0. If K is negative (mortality greater than the birth rate), only a zero population Xq = 0 is stable. It is Ae domain of extinction of the populations. If K is positive, two solutions exist an extinguished population Xo = 0 and a stationary population Xo = K. The stability analysis shows that the solution Xo = 0 becomes xmstable whereas the stable situation corresponds to Xo = K. The point K = 0 is thus a bifurcation point characteristic of a deep change in the nature of the mathematical solutions. 

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