Hopf bifurcation points

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... 

The stability of the (lAe)-family is lost at a Hopf bifurcation point denoted by the open circle (o) on Fig. 7, where the real parts of a complex conjugate pair of eigenvalues change sign. No stable time-periodic solutions were found near this point, indicating that the time-periodic states evolve sub-critically in P and are unstable. Haug (1986) predicted Hopf bifurcations for codimension two bifurcations of the form shown in Fig. 7. but did not compute the stability of the time-periodic states. 

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. 

Imaginary, (real parts = 0) Hopf bifurcation point or centre... 

Fig. 3.6. The dependence of local stability and character of the stationary-state solution on the parameters /i amd ku. (a) The locus of Hopf bifurcation points with tr(J) = 0 beneath...

Fig. 3.7. Stationary-state loci a( ) and / (/<) showing changes in local stability when the uncatalysed step is included in the model and ku <, showing two Hopf bifurcation points H and n. Particular numerical values correspond to parameter values in Table 3.1.

Before we can conclude, in general, that a given system will begin to show oscillatory behaviour between two Hopf bifurcation points we must attend to a few additional requirements of the theorem. 

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. 

As the dimensionless concentration of the reactant decreases so that pi just passes through the upper Hopf bifurcation point pi in Fig. 3.8, so a stable limit cycle appears in the phase plane to surround what is now an unstable stationary state. Exactly at the bifurcation point, the limit cycle has zero size. The corresponding oscillations have zero amplitude but are born with a finite period. The limit cycle and the amplitude grow smoothly as pi is decreased. Just below the bifurcation, the oscillations are essentially sinusoidal. The amplitude continues to increase, as does the period, as pi decreases further, but eventually attains a maximum somewhere within the range pi% < pi < pi. As pi approaches the lower bifurcation point /zf from above, the oscillations decrease in size and period. The amplitude falls to zero at this lower bifurcation point, but the period remains non-zero. 

Because oscillatory behaviour persists only for a finite length of time, only a finite number of excursions can occur. We can estimate this number by obtaining an approximate value for the mean oscillatory period, im. For this we take a geometric mean of the periods at the two Hopf bifurcation points. These latter quantities can be evaluated from the frequency co0 defined by... 

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... 

There are then, also, two values of the dimensionless concentration of the reactant, /if and /if with /if > /if say, on this locus. For our example these are /if = 0.05797 and /if = 0.2070. In between these solutions, the stationary state is unstable. For any other particular system with a different value of k, the appropriate Hopf bifurcation points can be calculated in a similar way, as given in Table 4.3, or read off Fig. 4.3. However, if k is small, we can also estimate /if and /if directly by using an approximate, but quite accurate,... 

We may also note from this last result that the lower Hopf bifurcation point /i f lies at a slightly higher value of /i than the maximum in the ass locus (which occurs at /i = k). 

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 stationary-state locus as /r varies (Fig. 4.4). If these bifurcation points are /r and /z (with the stationary state... 

The emerging limit cycle is born when the dimensionless reactant concentration has the value fx the cycle grows as n then varies away from n. There are two possibilities the limit cycle can grow as fx increases, i.e. for n > n, or as ix decreases, with n < n. Which of these two applies at any given bifurcation point is determined by the sign of a parameter ix2 (we retain the conventional notation for this quantity at the slight risk of confusion between this and the value of the dimensionless reactant concentration at the lower Hopf bifurcation point, fi ). The appropriate form for /x2 for the present model is... 

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. 

For the upper Hopf bifurcation point, with n >2, the stability exponent / 2 is negative as is the term p2. The first fact (/ 2 < 0) means that the limit cycle emerging from the bifurcation is stable. The particular sign of p2 means that the limit cycle grows as p is decreased below p, as shown in Fig. 4.4. 

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. 

Example applications of the Hopf formulae for pool chemical model with exponential approximation. The two sets of data for each k correspond to lower and upper Hopf bifurcation points respectively... 

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... 

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...

Location of degenerate Hopf bifurcation points from eqn (5.55)... 

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. ... 

Fig. 5.6. Typical tc-g parameter plane for the thermokinetic model with the full Arrhenius temperature depedence a region of stationary-state instability lies within the locus of Hopf bifurcation points (solid curve). Also shown, as broken lines, are the loci corresponding to the maximum and minimum in the g(a, 0) = 0 nullcline (see text).

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). 

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). 

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