Hopf bifurcation analyses

To save continual referencing back to the previous chapter we collect here the important equations for the scheme P - A - B + heat. The governing rate equations are 

The eigenvalues and X2 which determine the local stability of the stationary state are given by the roots of the equation 

The conditions for Hopf bifurcation require the trace to become zero and can be expressed parametrically as 

The locus described by these equations in the K-fx parameter plane is reproduced in Fig. 5.1, which also shows a typical stationary-state bifurcation diagram for fixed k. Hopf bifurcations occur at two values of the precursor reactant concentration /if, 2, with /if /if, for any given k less than a 

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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... 

Hopf bifurcation analysis commonly signals the onset of oscillatory behaviour. This chapter uses a particular two-variable example to illustrate the essential features of the approach and to explore the relationship to relaxation oscillations. After a careful study of this chapter the reader should be able to ... 

In this chapter we give an introduction and recipe for the full Hopf bifurcation analysis for chemical systems. Rather than work in completely general and abstract terms, we will illustrate the various stages by using the thermokinetic model of the previous chapter, with the exponential approximation for simplicity. We can draw many quantitative conclusions about the oscillatory solutions in that model. In particular we will be able to show (i)that the parameter values given by eqns (4.49) and (4.50) for tr(J) = 0 satisfy all the requirements of the. Hopf theorem (ii)that oscillatory behaviour is completely confined to the conditions for which the stationary state is... 

Hopf bifurcation analysis with Arrhenius model birth and growth of oscillations... 

We have already discussed the expressions resulting from a full Hopf bifurcation analysis of the thermokinetic model with the exponential approximation (y = 0). We may do the same for the exact. Arrhenius temperature dependence (y 0). Although the algebra is somewhat more onerous, we still arrive at analytical, expressions for the stability of the emerging or vanishing limit cycle and the rate of growth of the amplitude and period at... 

The Hopf bifurcation analysis proceeds as described previously, the required condition being that the trace of the Jacobian matrix corresponding to eqns (12.45) and (12.46) should become equal to zero for some stationary-state concentration given by the lower root from (12.51). (The solution with the upper root corresponds to the middle branch of stationary states for... 

Olsen, R. J. Epstein, I. R. Bifurcation analysis of chemical reaction mechanisms. 1. Steady-state bifurcation structure. J. Chem. Phys. 1991, 94, 3083-3095 Bifurcation analysis of chemical reaction mechanisms. 2. Hopf bifurcation analysis. J. Chem. Phys. 1993, 98, 2805-2822. 

The package does not do Hopf bifurcation analysis nor have any direct way to distinguishing between limit cycle and chaotic attractors. The package contains the Zero Deficiency Theorem, the "knot tree network theorem" as well as some older theorems that identify stable networks. The package solves the general reaction balancing problem whose solution is a convex polyhedral cone of extreme reactions. It handles thermodynamic properties of reactions assuming ideality. 

Depending on parameter values, there could be multiple states and unique unstable states. Some of the former and all of the latter would lead to sustained oscillations. The usual mathematical methods have been employed in the analysis of oscillatory behavior, including linear stability analysis, Hopf bifurcation analysis and computer simulations. 

From a mathematical point of view, the onset of sustained oscillations generally corresponds to the passage through a Hopf bifurcation point [19] For a critical value of a control parameter, the steady state becomes unstable as a focus. Before the bifurcation point, the system displays damped oscillations and eventually reaches the steady state, which is a stable focus. Beyond the bifurcation point, a stable solution arises in the form of a small-amplitude limit cycle surrounding the unstable steady state [15, 17]. By reason of their stability or regularity, most biological rhythms correspond to oscillations of the limit cycle type rather than to Lotka-Volterra oscillations. Such is the case for the periodic phenomena in biochemical and cellular systems discussed in this chapter. The phase plane analysis of two-variable models indicates that the oscillatory dynamics of neurons also corresponds to the evolution toward a limit cycle [20]. A similar evolution is predicted [21] by models for predator-prey interactions in ecology. 

To this end, address the equation (6.2.7) with estable limit cycle appears around the equilibrium point, i.e., a Hopf bifurcation takes place. We wish to study the solution that arises in the close vicinity of the bifurcation. Introduce a new time... 

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. 

In the next few sections we will concentrate on the form of the governing equations (4.24) and (4.25) with the exponential approximation to f(0) as given by (4.27). We will determine the stationary-state solution and its dependence on the parameters fi and k, the changes which occur in the local stability, and the conditions for Hopf bifurcation. Then we shall go on and use the full power of the Hopf analysis, to which we alluded in the previous chapter, to obtain expressions for the growth in amplitude and period of the emerging oscillatory solutions. 

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... 

Using the usual Hopf analysis we can readily find that the two-variable scheme, with fixed c, has a Hopf bifurcation at c = 1.102. Now let us consider the behaviour of the full three-variable scheme. 

The context of Uppal, Ray, and Poore s work was the revival of interest in Hopf bifurcation at the end of the 1960s and the soon-to-be-transmogrified catastrophe theory of Zeeman. Ray has followed up this type of analysis in studies of polymerization and other important processes. 

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. 

The discussion following Eqn. (5.1.8) imply a single Hopf bifurcation when Reynolds number increases beyond Rccr It is interesting to note that Landau (1944) talked about further instabilities following the nonlinear saturation of the primary instability mode. This is akin to Floquet analysis of the resulting time periodic system (Bender Orszag (1978)). The possibility of multiple bifurcation was also mentioned in Drazin Reid (1981) who stated that in more complete models of hydrodynamic stability we shall see that there may he further bifurcations from the solution A = 0, e.g. where the next least stable mode of the basic flow becomes unstable, and from the solution A = Ae- To the knowledge of the present authors, no theoretical analysis exist that showed multiple bifurcation before for this flow. Here,... 

The empty-site requirement in Eq. (28) can be physically interpreted in one of two different ways either the adsorbed A and B have to rearrange prior to reaction, or they are bound to more than one adsorption site. For the latter case, the intermediate concentration is low, thus allowing a pseudo-steady-state assumption. Through the application of bifurcation analysis and catastrophe theory this model was found to predict a very rich bifurcation and dynamic behavior. For certain parameter values, sub- and supercritical Hopf bifurcations as well as homoclinic bifurcations were discovered with this simple model. The oscillation cycle predicted by such a model is sketched in Fig. 6c. This model was also used to analyze how white noise would affect the behavior of an oscillatory reaction system... 

Limit cycles also appear in Chapter 4, but no bifurcation theorems were used (although the Hopf bifurcation theorem could have been used). Uniqueness of these cycles is a question of major interest and importance. To more accurately model the chemostat as it is used in commercial production, the plasmid model discussed in Chapter 10 should be combined with the inhibitor model of Chapter 4. More specifically, consider two organisms - differing only by the presence or absence of a plasmid that confers immunity to the inhibitor - competing in a chemostat (equation (4.2) of Chapter 9 modified for the presence of the inhibitor). The techniques of analysis used in Chapter 4 do not apply, since the system is not competitive in the mathematical sense. Yet an understanding of this system would be very important. 

We begin the analysis of (4), (5) by constructing a trapping region and applying the Poincare-Bendixson theorem. Then we ll show that the chemical oscillations arise from a supercritical Hopf bifurcation. 

The article by Odell (1980) is worth looking up. It is an outstanding pedagogical introduction to the Hopf bifurcation and phase plane analysis in general. 

Odell, G. M. (1980) Qualitative theory of systems of ordinary differential equations, including phase plane analysis and the use of the Hopf bifurcation theorem. Appendix A.3. In L. A. Segel, ed.. Mathematical Models in Molecular and Cellular Biology (Cambridge University Press, Cambridge, England). 

Such an analysis of the Hopf bifurcation (and some other catastrophes) will be presented in Section 5.6. 

The way of presentation of the material discussed in Chapter 5 is based on papers of Guckenheimer and on the ideas contained in papers by Stewart. A paper by Nicolis and an article by Othmer in a book published by Field and Burger constitute a very good supplement to these papers. A book by Arnol d (1983), although rather difficult, provides much additional material. The elementary method of analysis of some dynamical catastrophes presented in Section 5.6 is patterned after ArnoPd s approach to the Hopf bifurcation in the van der Pol system described in his book (1975). A book by Gilmore provides basic information on catastrophes in dynamical systems. A paper by Stewart contains another proof (compared to Section 5.5) that Hopf bifurcation is an elementary catastrophy. 

On the basis of the above analysis we arrive at a conclusion that in the system represented by the equations not accounting for diffusion (6.82) the Hopf bifurcation may appear. We shall examine the catastrophe taking place in more detail using the method described in Section 5.6. 

From the above analysis a conclusion can be drawn that, because the stationary state (x2, y2, z2) can have a sensitive state corresponding at most to the Hopf bifurcation, the centre manifold theorem may be applied to the system of equations (6.98) thus reducing it to a system of equations in two variables in which the Hopf bifurcation would appear. 

It can be shown that there is a surface of Hopf bifurcations in the parameter space (ei, 2, /), so that oscillations in the chemical concentrations are expected at one side of that surface. Nevertheless, analysis is much simplified for interesting experimental situations leading to particular parameter values. To be specific we consider the situation analyzed in Scott (1991), for which 2 4 x 10-4 time derivative in Eq. (3.53) is very large except when the variables sit on the nullcline, —qy — xy + fz = 0. This makes y to grow or to decrease fast until the nullcline is reached (more precisely, a neighborhood of size 0(e 1) of the nullcline). After this moment... 

Comparing with section 3.2.4 we see how the tip dynamics (3.79) takes on the skew product form (3.12). Indeed we only have to assume the variable v to be T-periodic, as Hopf bifurcation had caused it to be there, and our current tip dynamics (3.79) fits right in. Defining tn = (Gkq), the analysis (3.20), (3.21) of meanders and drifts applies with... 

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