Hopf bifurcation condition

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 

2T( + K2tres) = 1 + (1 — 47 )1/2. This condition is satisfied when T = k212, so that 

In fact only the upper root corresponds to a Hopf bifurcation point (the lower solution to the condition tr(J) = 0 being satisfied along the saddle point branch of the isola where the system does not have complex eigenvalues). 

We can see from eqn (8.42) that k2 must be small if there are to be real solutions for the Hopf point. In fact the condition k212 i is exactly the same as that for the existence of isolas (k2 iV)- Thus all isolas have a point of Hopf bifurcation along their upper branch. 

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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 order to evaluate P2 we need to consider how the governing equations for mass and energy balance themselves vary with changes in the variables. In the case of the present model this means evaluating various partial derivatives of (5.1) and (5.2) with respect to a and 0. Before proceeding, however, we should take a look at the elements of the Jacobian matrix evaluated for Hopf bifurcation conditions ... 

From the results presented in this chapter, more advanced studies from the bifurcation theory can be planed. For example, inside the lobe, the behavior of the reactor is self-oscillating, i.e. an Andronov-Poincare-Hopf bifurcation can be researched from the calculation of the first Lyapunov value, in order to know if a weak focus may appear, or the conditions which give a Bogdanov-Takens bifurcation etc. Finally, it is interesting to remark that the previously analyzed phenomena should be known by the control engineer in order to either avoid them or use them, depending on the process type. 

We shall not specify here the initial conditions on C(x,t) since in what follows we shall only be preoccupied with the limit state resulting from a Hopf bifurcation from the following stationary solution of the above system... 

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... 

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 locus of these Hopf bifurcation points is also shown in Fig. 4.3 and can be seen to be another closed loop emanating from the origin. It lies in the region between the loci for changes between nodal and focal character, so the condition tr(J) separates stable focus from unstable focus. The curve has a maximum at... 

Conditions for Hopf bifurcation with exponential approximation... 

We have already determined the following information about the behaviour of the pool chemical model with the exponential approximation. There is a unique stationary-state solution for ass, the concentration of the intermediate A, and 0SS, the temperature rise, for any given combination of the experimental conditions /r and k. If the dimensionless reaction rate constant k is larger than the value e-2, then the stationary state is always stable. If heat transfer is more efficient, so that k Hopf bifurcation points along the stationary-state locus as /r varies (Fig. 4.4). If these bifurcation points are /r and /z (with the stationary state... 

Some typical oscillatory records are shown in Fig. 4.6. For conditions close to the Hopf bifurcation points the excursions are almost sinusoidal, but this simple shape becomes distorted as the oscillations grow. For all cases shown in Fig. 4.6, the oscillations will last indefinitely as we have ignored the effects of reactant consumption by holding /i constant. We can use these computations to construct the full envelope of the limit cycle in /r-a-0 phase space, which will have a similar form to that shown in Fig. 2.7 for the previous autocatalytic model. As in that chapter, we can think of the time-dependent... 

The condition for Hopf bifurcation, i.e. for a change from stable to unstable focus, is also shown in Fig. 4.8. This is given parametrically by... 

Equation (4.74) has distinct real roots provided y < . Hopf bifurcation cannot occur if the activation energy E becomes too small compared with the thermal energy RT i.e. if E < 4RTa. This is the same condition on y as that for the existence of the maximum and minimum in the ass locus. In fact, the Hopf bifurcation points always occur for p values between the maximum and minimum, i.e. on the part of the locus where ass is decreasing, as shown in Fig. 4.8(b) where the loci of turning points are shown as broken lines. 

First, can we expect any oscillatory behaviour Instability is possible only if k < e 2. This requirement is satisfied here. From the data in Table 4.4, the Hopf bifurcation points for this system occur for n = 0.207 and n = 0 058. For our example, the initial value /r0 = 0.5 exceeds the upper bifurcation point, so the system at first has a stable pseudo-stationary state to approach, with dss x 10 and ass x 4.54 x 10 4. From Fig. 4.3 we may also estimate that the approach to this state will be monotonic since the initial conditions lie outside the region of damped oscillations. 

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... 

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

First, we recall the conditions for Hopf bifurcation derived in 4.9.2(b). These are... 

We also have the hint of a new type of degeneracy associated with systems possessing multiple stationary states. It is possible for both the trace and the determinant of the Jacobian matrix to become zero simultaneously this gives the system two eigenvalues which are both equal to zero. These double-zero eigenvalue situations are important because they represent conditions at which a Hopf bifurcation point with an associated homoclinic orbit first appears. In the present case, tr(J) = det(J) = 0 only when k2 = Vg, but then the isola has shrunk to a point. 

The condition tr(J) = det(J) = 0 corresponds to a Hopf bifurcation point moving exactly onto the saddle-node turning point (ignition or extinction point) on the stationary-state locus. Above the curve A the system may have two Hopf bifurcations, or it may have none as we will see in the next subsection. Below A there are two points at which tr (J) = 0, but only one of... 

Fig. 8.11. The locus H of degenerate Hopf bifurcation points described by the transversality condition (merging of two Hopf points), eqn (8.51). Below this curve, the stationary-state locus exhibits Hopf bifurcation (dynamic instability) at some residence times above it, the system does...

Of particular interest is the special case of a complex pair of principal eigenvalues whose real parts are passing through zero. This is the situation which we have seen corresponding to a Hopf bifurcation in the well-stirred systems examined previously. Hopf bifurcation points locate the conditions for the emergence of limit cycles. Using the CSTR behaviour as a guide it is relatively easy to find conditions for Hopf bifurcations, and then locally values of the diffusion coefficient for which a unique stationary state is unstable. Indeed the stationary-state profile shown in Fig. 9.5 is such a... 

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.

Fig. 10.7. Representation of conditions in the K-fi parameter plane for instability of the uniform stationary state with respect to spatial perturbations for a system with f = 10. Also shown (broken curve) is the Hopf bifurcation locus, within which the uniform state is itself unstable. The two loci cross at some point on their upper shores.

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... 

We have specified that the conditions of interest here are those lying within the Hopf bifurcation locus, so tr(U) will be greater than zero. This means that the eigenvalues appropriate to this uniform state must have positive real parts. The system is unstable to uniform perturbations. We must exclude this n = 0 mode from any perturbations in the remainder of this section. 

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... 

The stationary state is unstable for p > p. Thus there will be a point of Hopf bifurcation provided the rate constants for reaction and the adsorption and desorption of the poison satisfy the condition... 

Again we have a two-variable system, so we can look for points of Hopf bifurcation in terms of the trace and determinant of the Jacobian matrix evaluated for the stationary-state solutions. Thus we seek conditions such that... 

As the parameter A is varied, so the point of intersection of the mapping curve and the straight line in Fig. 13.5 changes. The gradient of the cubic curve at the point of intersection also varies during this process. It turns out that the stationary state represented by the intersection is stable provided the value of the gradient at the point is greater than - 1. The condition for the equivalent of a Hopf bifurcation in the map is thus... 

As described above, the first condition on the eigenvalues for a Hopf bifurcation in a three-variable scheme is that the principal pair should be purely imaginary and the third should be real and negative. For this to be the... 

The Hopf bifurcation where a stable focus becomes unstable and sheds or absorbs a periodic solution is an important transition which has received a great deal of attention (for a review see Marsden McCracken 1976). Clearly the lines over which it can take place are the loci of steady-states whose eigenvalues are purely imaginary. These are shown on the sides of the fin in figure 5. Because this is a two-dimensional system we can write down the condition quite explicitly. Writing the equations ... 

The point S of figure 8 at which the Hopf bifurcation curve crosses the boundary of the multiplicity region is not a double zero degeneracy, for the upper steady state (i.e. that with the larger 0b) is undergoing the Hopf bifurcation at the same time as the lower steady-state undergoes a saddle-node bufurcation, i.e. the conditions trJ = 0 and detJ = 0 apply at different points. It does, however, serve to show the four combinations of the two most common... 

Figure 12.6a shows the temporal variation of the proximal tubular pressure Pt as obtained from the single-nephron model for a = 12 and T = 16 s. All other parameters attain their standard values as listed in Table 12.1. Under these conditions the system operates slightly beyond the Hopf bifurcation point, and the depicted pressure variations represent the steady-state limit cycle oscillations reached after the initial transient has died out For physiologically realistic parameter values the model reproduces the observed self-sustained oscillations with characteristic periods of 30-40 s. The amplitudes in the pressure variation also correspond to experimentally observed values. Figure 12.6b shows the phase plot Here, we have displayed the normalized arteriolar radius r against the proximal intratubular pressure. Again, the amplitude in the variations of r appears reasonable. The motion... 

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