Limit cycles Hopf bifurcation

Figure C3.6.5 The first two periodic orbits in the main subhannonic sequence are shown projected onto the (c, C2) plane. This sequence arises from a Hopf bifurcation of the stable fixed point for the parameters given in the text. The arrows indicate the direction of motion, (a) The limit cycle or period-1 orbit at k 2 = 0.11. (b) The first subhannonic or period-2 orbit at k 2 = 0.095.

The quasiperiodic route to chaos is historically important. It arises from a succession of Hopf birfurcations. As already noted, a single Hopf bifurcation results in a limit cycle. The next Hopf bifurcation produces a phase flow tliat can be represented on tire surface of a toms (douglmut). This flow is associated witli two frequencies if tire ratio of tliese frequencies is irrational tlien tire toms surface is densely covered by tire phase trajectory, whereas if... 

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. 

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. 

Recall that a Hopf bifurcation is termed supercritical if its bifurcation diagram is as shown schematically in Fig. 6.2.2a. Correspondingly, in this case a stable limit cycle is born around the equilibrium, unstable hereon, only at a critical (bifurcation) value of the control parameter A = Ac. In contrast, in the subcritical case (Fig. 6.2.2b), the equilibrium is surrounded by limit cycles already for A < Ac, with an unstable limit cycle separating the stable one from the still stable equilibrium. At the bifurcation A = Ac the unstable limit cycle dies out with the equilibrium, unstable hereon, surrounded by a stable limit cycle. Thus the main feature of the subcritical case (as opposed to the supercritical one) is that a stable equilibrium and a stable limit cycle coexist in a certain parameter range, with a possibility to reach the limit cycle through a sufficiently strong perturbation of the equilibrium. 

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 must also examine the stability of the periodic solution and its limit cycle as it emerges from the bifurcation point. Just as stationary states may be stable or unstable, so may oscillatory solutions. If they are stable they may be observable in practice if they are unstable they will not be directly observable although their existence still has some physical relevance. We will give the recipe for evaluating the stability and character of a Hopf bifurcation in the... 

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. 

There are no unstable limit cycles in this model, and the oscillatory solution born at one bifurcation point exists over the whole range of stationary-state instability, disappearing again at the other Hopf bifurcation. Both bifurcations have the same character (stable limit cycle emerging from zero amplitude), although they are mirror images, and are called supercritical Hopf bifurcations. 

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... 

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. 

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 various Hopf bifurcation parameters / 2, p2, and t2 can again be determined explicitly but have much more complex forms. We will discuss the details in the next chapter and only consider here the stability of the emerging limit cycle through P2. With the full Arrhenius form, this parameter is given by... 

Fig. 4.9. The development of oscillatory amplitude Ae and period T across the range of instability, 4.2 x 10 3 = n < n < jx = 0.0195, for the pool chemical model with k = 2x 10-3 and y = 0.21, typical of a system with a subcritical Hopf bifurcation at which an unstable limit cycle emerges at The broken curves give the limiting forms predicted by eqns (4.59)—(4.61).

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. 

At the point of Hopf bifurcation, the emerging limit cycle has zero amplitude and an oscillatory period given by 2n/a>0. As we begin to move away from the bifurcation point the amplitude A and period T grow in a form we can calculate according to the formulae... 

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... 

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...

Table 5.1 gives the location of the / 2 = 0 point for various values of y. Points such as this, where the nature of the limit cycle changes qualitatively, are known as degenerate Hopf bifurcations . 

At the lower Hopf bifurcation, P2 is always negative and n2 is positive. Thus a stable limit cycle emerges from n, growing as the reactant concentration jx is increased. 

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. ... 

If k2 is greater than ys, we know there will be no isola and no Hopf bifurcation point. For k2 < /g, but greater than 9/256, P2 is positive. This means that the emerging limit cycle will be unstable. The limit cycle grows as the residence time is reduced below the bifurcation point t s surrounding the upper stationary state which is stable. 

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. 

Fig. 8.6. Typical arrangement of local stabilities and development of unstable limit cycle, from a subcritical Hopf bifurcation, appropriate to cubic autocatalysis with decay and no autocatalyst inflow and with 9/256 < k2 < 1/16. The unstable limit cycle grows as t, decreases below t m, and terminates by means of the formation of a homoclinic orbit at rf . Stable stationary states, including the zero conversion branch 1 — a, = 0, are indicated by solid curves, unstable states and limit cycles by broken curves.

We should also consider the behaviour along the top of the isola, on the part of the branch lying at longer residence times than the Hopf point. For Tres > t s, and with k2 still in the above range, the uppermost stationary state is unstable and is not surrounded by a stable limit cycle. The system cannot sit on this part of the branch, so it must eventually move to the only stable state, that of no conversion. Thus we fall off the top of the isola not at the long residence time turning point, but earlier as we pass the Hopf bifurcation point. 

Fig. 8.7. Supercritical Hopf bifurcation for cubic autocatalysis with decay and /) = 0, appropriate for small dimensionless decay rate constant k2 < 9/256. A stable limit cycle emerges and grows as the residence time is increased above t s. At higher residence times, this disappears at rj , by merging with an unstable limit cycle born from a homoclinic orbit at t. (With non-zero autocatalyst inflow, (i0 > 0, the stable limit cycle itself may form a homoclinic orbit at long tres.)...

Fig. 8.8. Phase plane representations of the birth (or death) of limit cycles through homoclinic orbit formation. In the sequence (a)-fb)-(c) the system has two stable stationary states (solid circles) and a saddle point. As some parameter is varied, the separatrices of the saddle join together to form a closed loop or homoclinic orbit (b) this loop develops as the parameter is varied further to shed an unstable limit cycle surrounding one of the stationary states. The sequence (d)-(e)-(f) shows the corresponding formation of a stable limit cycle which surrounds an unstable stationary state. (In each sequence, the limit cycle may ultimately shrink on to the stationary state it surrounds—at a Hopf bifurcation point.)...

These requirements specify two loci one of them, labelled DH l in Fig. 8.12, emanates from the points / = 0, k2 = 9/256, as located in 8.3.6. This curve cuts through the parameter space for isola and mushroom patterns, but always lies below the curve A. (In fact it intersects A at the common point P0 = i(33/2 - 5), k2 = rg(3 - /3)4(1 -, /3)2 where the locus H also crosses.) In the vicinity of DH x, the stationary-state curve has only one Hopf point. This changes from a subcritical bifurcation (unstable limit cycle emerging) for conditions to the right of the curve to supercritical (stable limit cycle emerging) to the left. 

Fig. 8.12. The loci DH, and DH2 corresponding to degenerate Hopf bifurcation points at which the stability of the emerging limit cycle is changing. Again, these are shown relative to the loci for stationary-state multiplicity (broken curves).

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