Stability condition bifurcation

If x =0, the previous condition only implies that l i trivial solution of extinct population as long as it is the only possible solution. When x 0, the same condition implies that p will not be greater than 3. The stability condition in this case is l[Pg.27]

A spatial Hopf bifurcation, commonly called a wave bifurcation, corresponds to a pair of purely imaginary eigenvalues for some 0> Ci = 0 and C2 > 0. According to the stability conditions (10.23), T < 0, and therefore... 

Since T is negative according to the stability conditions (10.23), Ci is always positive. The positivity of the Hurwitz determinant A2 and C4 at the Hopf bifurcation imply that C3 > 0 there and that well defined [205]. Gathering terms of equal powers in k, we rewrite the Hopf condition as... 

Note that the third stability condition corresponds to Aj > 0. According to (1.38), the wavenumber zero mode of the activator-inhibitor-substrate system undergoes a Hopf bifurcation if A2 goes through 0. For mass-action kinetics, (12.20) implies that G or Po- Consequently, the third stability condition can fail, if the total substrate is too low. Then the Turing bifurcation ceases to be the primary instability, and a uniform Hopf bifurcation occurs first in the fiiU system. 

The coefficients C K) of the characteristic polynomial det[J( )—X tl4] = 0 and the Hurwitz determinants A K) are easily obtained using computational algebra software such as Mathematic A (Wolfram Research, Inc., Champaign, IL) or Maple (Waterloo Maple Inc., Waterloo, Ontario). The th mode undergoes a stationary bifurcation when condition (12.41d) is violated, namely c K) = 0, as discussed in Sect. 1.2.3, see (1.36). In other words, a Turing bifurcation of the uniform steady state corresponds to c iki) = 0 with k 0, while the stability conditions (12.41) are satisfied for all other modes with k fej. The feth mode undergoes an oscillatory bifurcation when condition (12.41c) is violated, namely A K) = 0, as discussed in Sect. 1.2.3, see (1.38). A wave bifurcation of the uniform steady state corresponds to A k i) = 0 with k f/ 0, while the stability conditions (12.41) are satisfied for all otha modes with k few As discussed in Sect. 10.1.2, see (10.29), a wave bifurcation cannot occur in a two-variable reaction-diffusion system. 

The above equation is similar to that obtained by Sri Namachchivaya and the results of this previous work can be applied provided R > 0. As indicated earlier for 6 > 0, which is generally the case when eigenvalues cross from left to right in the complex x-plane, the stable bifurcation path exists for n > 0. Thus, we assume the deterministic system exhibits a supercritical Hopf bifurcation throughout this work. It is evident from (15) that the stability condition for the n moment of the linear system can be written as... 

Figure 11 Relationship between structure bifurcation and stability condition in gas-solid systems. Reprinted from Chen et al (2012) with permission from Elsevier.

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. 

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. 

For all physically acceptable conditions, the determinant of J is positive, so we will not find saddle points or saddle-node bifurcations. We can, however, expect to find conditions under which nodal states become focal (damped oscillatory responses), i.e. where A = 0, and where focal states lose stability at Hopf bifurcations, i.e. where tr(J) = 0 and where we shall look for the onset of sustained oscillations. 

The condition for a change in the local stability of the stationary state in this model is that the trace of the Jacobian matrix should be zero. We can also recognize this as the first requirement for Hopf bifurcation, about which we shall have more to say in the next section. The condition tr(J) = 0 is also most easily handled parametrically by replacing n by k0 wherever possible in eqn (4.42). This leads to... 

The condition for a change in stability along the uppermost branch of solutions, by means of a Hopf bifurcation, is tr(J) = 0. This generally occurs as the residence time is increased, when... 

Fig. 10.3. The locus of Hopf bifurcation points indicating the conditions for loss of local stability for the spatially uniform stationary-state solution. Inside this region the system may show spatially uniform time-dependent oscillations.

In the previous section we have taken care to keep well away from parameter values /i and k for which the uniform stationary state is unstable to Hopf bifurcations. Thus, instabilities have been induced solely by the inequality of the diffusivities. We now wish to look at a different problem and ask whether diffusion processes can have a stabilizing effect. We will be interested in conditions where the uniform state shows time-dependent periodic oscillations, i.e. for which /i and k lie inside the Hopf locus. We wish to see whether, as an alternative to uniform oscillations, the system can move on to a time-independent, stable, but spatially non-uniform, pattern. In fact the... 

There is a voluminous literature on steady-state multiplicity, oscillations (and chaos), and derivation of bifurcation points that define the conditions that lead to onset of these phenomena. For example, see Morbidelli et al. [ Reactor Steady-State Multiplicity and Stability, in Chemical Reaction and Reactor Engineering, Carberry and Varrria (eds), Marcel Dekker, 1987], Luss [ Steady State Multiplicity and Uniqueness... 

For both analyses, the procedure is to use dimensionless parameters in the set of differential equations describing the model, look for the steady state, investigate the linear stability, and determine the conditions for instability. Near the bifurcation values of the parameters, which initiate an oscillatory growing solution, a perturbation analysis provides an estimate for the period of the ensuing limit cycle behavior. 

Thus, an increase in the value of the controlling parameter a can result either in the violation of the condition of aperiodicity in relaxation pro cesses at the preserved total stabihty of the system or, vice versa, in the vio lation of the condition of stationary state stability by which the system is transferred to the nonthermodynamic branch. It is important that the properties of the system state change jumpwise when passing through the bifurcation point, and thus these changes are called sometimes the kinetic phase transitions. 

The behaviour at the upper Hopf point is also that of a supercritical Hopf bifurcation although the loss of stability of the steady-state and the smooth growth of the stable limit cycle now occurs as the parameter is reduced. This is sketched in Fig. 5.10(b). We can join up the two ends of the limit cycle amplitude curve in the case of this simple Salnikov model to show that the amplitude of the limit cycle varies smoothly across the range of steady-state instability, as indicated in Fig. 5.11(a). The limit cycle born at one Hopf point survives across the whole range and dies at the other. Although this is the simplest possibility, it is not the only one. Under some conditions, even for only very minor elaboration on the Salnikov model [16b], we encounter a subcritical Hopf bifurcation. At such an event, the limit cycle that is born is not stable but is unstable. It still has the form of a closed loop in the phase plane but the trajectories wind away from it, perhaps back in towards the steady-state as indicated in Fig. 

---