The Hopf Bifurcation

Assume also that x (a) is a critical point of Eq. (29). The eigenvalues Pi( ) Piia). pjv(a) will now also depend on the parameter a. If for some values of a, say a Gq, the critical point is stable, and if in addition a pair of complex conjugate eigenvalues pi(a), p2(o) cross the imaginary axes transversely (d Repi(a)/da a=ao5 0) then we say that a Hopf bifurcation takes place at the value a = Gq. If a Hopf bifurcation occurs in Eq. (29) then there exists a one-parameter family of periodic solutions for a in the neighborhood of ao with a period near 27r/ Im pi(ao)l- K the flow attracts to the critical point x (ao) when a = Gq, then x°(ao) is called a vague attractor. For this case the family of closed orbits is contained m a Go and the orbits are of attracting type.  

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This complex Ginzburg-Landau equation describes the space and time variations of the amplitude A on long distance and time scales detennined by the parameter distance from the Hopf bifurcation point. The parameters a and (5 can be detennined from a knowledge of the parameter set p and the diffusion coefficients of the reaction-diffusion equation. For example, for the FitzHugh-Nagumo equation we have a = (D - P... 

One may also observe a transition to a type of defect-mediated turbulence in this Turing system (see figure C3.6.12 (b). Here the defects divide the system into domains of spots and stripes. The defects move erratically and lead to a turbulent state characterized by exponential decay of correlations [59]. Turing bifurcations can interact with the Hopf bifurcations discussed above to give rise to very complicated spatio-temporal patterns [63, 64]. 

This route should already be familiar to us from our discussion of the logistic map in chapter 4, Prom that chapter, we recall that the Feigenbaum route calls for a sequence of period-doubling bifurcations pitchfork bifurcations versus the Hopf bifurcations of the Landau-Hopf route) such that if subharmonic bifurcations are observed at Reynolds numbers TZi and 7 2, another can be expected at TZ determined by... 

THE HOPF BIFURCATION OR THE CHANGING NATURE OF EQUILIBRIUM POINTS PROBLEM OP WALAS... 

Since the only equilibrium point E(0 po) in the phase plane becomes unstable for i > ic and the infinity is unstable for any i, we conclude that limit cycles must exist around E(G,p0) for > ic. At the same time, the proven nonexistence of the limit cycles for i < ic implies the supercritical nature of the Hopf bifurcation at = ic in the symmetric case /"(0) = 0. 

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... 

The question of what happens to the system in the range of instability, and how the concentrations of A and B vary as they move away from the unstable stationary state, leads us to the study of sustained oscillatory behaviour. Before a full appreciation of the latter can be obtained, however, we must rehearse the relevant theoretical background. Fortunately the autocatalytic model is again an exemplary system with which to introduce at least the basic aspects of the Hopf bifurcation, and we will do this in the next section. 

The size and period of the oscillations, or of the corresponding limit cycle, varies with the dimensionless reactant concentration pi. We may determine this dependence in a similar way to that used in 2.5. Close to the Hopf bifurcation points we can in fact determine the growth analytically, but in general we must employ numerical computation. For now we will merely present the basic result for the present model. The qualitative pattern of response is the same for all values of ku < g. 

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 form only applies close to the Hopf bifurcation point, but it is here that numerical methods such as direct integration of the equations converge most slowly. 

Table 4.4 illustrates the application of the above formulae for systems with a range of k values. The Hopf bifurcation points are located by solving eqns (4.49) and (4.50) for a given k. 

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... 

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. 

Fig. 5.1. (a) A typical parameter plane showing a locus of Hopf bifurcation points. For any given value of the parameter k on the ordinate we may construct a horizontal (broken line) the Hopf bifurcation points, pi and pi, are then located as shown. The corresponding stationary-state loci, shown in (b) and (c), have unstable solutions between pi and pi. ... 

We have stressed that both the real and imaginary parts depend on the parameter n because we are imagining experiments where the reactant concentration will be varied whilst k is held constant. If we were doing the experiments another way so that n was held fixed and the dimensionless reaction rate constant varied in the vicinity of the Hopf bifurcation point we would then wish to consider v(/c) and a>(/c). 

Another derivative evaluated at the Hopf bifurcation point of interest, which we will need later on, is that of the imaginary part of frequency (d[Pg.115]

When we come to look at the stability of the limit cycle which is born at the Hopf bifurcation point, we shall meet a quantity known as the Floquet multiplier , conventionally denoted p2, which plays a role similar to that played for the stationary state by the eigenvalues and k2. If / 2 is negative, the limit cycle will be stable and should correspond to observable oscillations if P2 is positive the limit cycle will be unstable. 

Only four of these partial derivatives are not identically zero. For these we may use the following relationships, which apply at the Hopf bifurcation point, to simplify... 

We must remember that these forms apply only at the bifurcation point, but clearly once we know x, which is the value of the dimensionless temperature rise dss, at the Hopf bifurcation, all the above terms can be evaluated easily.)... 

Note that this involves the derivative of the real parts -of the eigenvalues Alt 2 or, equivalently, of the trace of J evaluated at the Hopf bifurcation point. We know that Reflj 2) is passing through zero at this point. 

For t2, therefore, we need the derivative of the imaginary part of the eigenvalues evaluated at the Hopf bifurcation point. We may also note that the sign of the quotient t2//r2 is of less immediate significance than those of / 2 and n2. 

Fig. 5.3. Locus of Hopf bifurcation points in K-fi parameter plane for thermokinetic model with the full Arrhenius temperature dependence and y = 0.21. The nature of the Hopf bifurcation point and, hence, the stability of the emerging limit cycle changes along this locus at k = 2.77 x 10 3. Supercritical bifurcations are denoted by the solid curve, subcritical bifurcations occur along the broken segment, i.e. at the upper bifurcation point for the lowest k. The stationary-state solution is unstable and surrounded by a stable limit cycle for all parameter values within the enclosed region. Oscillatory behaviour also occurs in the small shaded region below the Hopf curve, where the stable stationary state is surrounded by both an unstable and...

When the Hopf bifurcation at p is supercritical (/ 2 < 0) the system has just a single stable limit cycle. This emerges at p and exists across the range p < p < p, within which it surrounds the unstable stationary-state solution. The limit cycle shrinks back to zero amplitude at the lower bifurcation point p%. This behaviour is qualitatively the same as that shown with the simplifying exponential approximation and is illustrated in Fig. 5.4(a). 

FlO. 5.4. The birth and growth of oscillatory solutions for the thermokinetic model with the full Arrhenius temperature dependence, (a) The Hopf bifurcations /x and ft are both supercritical, with [12 < 0, and the stable limit cycle born at one dies at the other, (b) The upper Hopf bifurcation is subcritical, with fl2 > 0. An unstable limit cycle emerges and grows as the dimensionless reactant concentration ft increases—at /rsu this merges with the stable limit cycle born at the lower supercritical Hopf bifurcation point ft. ... 

The Hopf bifurcation approach is a mathematically rigorous technique for locating and analysing the onset of oscillatory behaviour in general dynamical systems. Another approach which has been particularly well exploited for chemical systems is that of looking for relaxation oscillations. Typically, the wave form for such a response can be broken down into distinct periods,... 

We can illustrate this latter technique with the simple thermokinetic model with the Arrhenius temperature dependence discussed above. This will also allow us to see that the two approaches are not separate, but that oscillations change smoothly from the basically sinusoidal waveform at the Hopf bifurcation to the relaxation form in other parts of the parameter plane. 

Thus we have an explicit formula in this case for the Hopf bifurcation points as a function of the decay rate constant for k2 = 20, Tres = 39.25 for k2 = Tres = 163.2. Figure 8.5 shows how the bifurcation point moves to longer residence times as k2 decreases, along with the locations of the extinction points t s from eqn (8.27). 

We now know that if a system on the upper branch of the isola, just below the Hopf bifurcation point, is given a small perturbation which remains within the unstable limit cycle, it will decay back to the upper solution. If, however, the perturbation is larger, so we move to a point outside the cycle, we will not be able to get back to the upper solution the system must move to the other stable state, with no reactant consumption. 

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