Stable limit cycle

The limit cycle O is an isolated closed trajectory having the property that dll other trajectories O in its neighborhood are certain spirals winding themselves onto C either for t - oo in which case C is called a stable limit cycle) or for t- - — ao (an unstable limit cycle). 

The reader can recognize that a stable limit cycle illustrates the definition (5) while a harmonic oscillator illustrates that given by (4). 

The opposite effect is depicted on Fig. 8.32 where the catalyst under open-circuit conditions exhibits stable limit cycle behaviour with a period of 184 s. Imposition of a negative current of -400 pA leads to a steady state. Upon current interruption the catalyst returns to its initial oscillatory state. Application of positive currents leads to higher frequency oscillatory states. 

Modify the program to generate a dimensionless, phase-plane display of BDIM versus ADIM, repeating the studies of Exercise 1. Note that the oscillatory behaviour tends to form stable limit cycles in which the average yield of B can be increased over steady-state operation. 

A two-variable model taking into account the allosteric (i.e. cooperative) nature of the enzyme and the autocatalytic regulation exerted by the product shows the occurrence of sustained oscillations. Beyond a critical parameter value, the steady state admitted by the system becomes unstable and the system evolves toward a stable limit cycle corresponding to periodic behavior. The model accounts for most experimental data, particularly the existence of a domain of substrate injection rates producing sustained oscillations, bounded by two critical values of this control parameter, and the decrease in period observed when the substrate input rate increases [31, 45, 46]. 

Yet another type of complex oscillatory behavior involves the coexistence of multiple attractors. Hard excitation refers to the coexistence of a stable steady state and a stable limit cycle—a situation that might occur in the case of circadian rhythm suppression discussed in Section VI. Two stable limit... 

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 eequilibrium 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... 

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. 

Fig. 3.8. Representation of the onset, growth, and death of oscillations in the isothermal autocatalytic model as /z varies for reaction with the uncatalysed step included, showing emergence of the stable limit cycle at and its disappearance at n. ...

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. 

Another useful rule which can frequently guide us to situations where oscillatory solutions will be found is the Poincare-Bendixson theorem. This states that if we have a unique stationary state which is unstable, or multiple stationary states all of which are unstable, but we also know that the concentrations etc. cannot run away to infinity or become negative, then there must be some other non-stationary atractor to which the solutions will tend. Basically this theorem says that the concentrations cannot just wander around for an infinite time in the finite region to which they are restricted they must end up somewhere. For two-variable systems, the only other type of attractor is a stable limit cycle. In the present case, therefore, we can say that the system must approach a stable limit cycle and its corresponding stable oscillatory solution for any value of fi for which the stationary state is unstable. 

This quick test does not, however, tell us that there will be only one stable limit cycle, or give any information about how the oscillatory solutions are born and grow, nor whether there can be oscillations under conditions where the stationary state is stable. We must also be careful in applying this theorem. If we consider the simplified version of our model, with no uncatalysed step, then we know that there is a unique unstable stationary state for all reactant concentrations such that /i < 1. However, if we integrate the mass-balance equations with /i = 0.9, say, we do not find limit cycle behaviour. Instead the concentration of B tends to zero and that for A become infinitely large (growing linearly with time). In fact for all values of fi less than 0.90032, the concentration of A becomes unbounded and so the Poincare-Bendixson theorem does not apply. 

Recalling that x is identically 9SS and hence always greater than or equal to unity, we see that the Floquet multiplier is always negative with this scheme only a stable limit cycle emerges. 

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...

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. ... 

Fig. 5.5. A typical phase portrait for a system with ft < ft < ft, showing a stable stationary-state solution (singular point) surrounded first by an unstable limit cycle (broken curve) and then by a stable limit cycle (solid curve). The unstable limit cycle separates those initial conditions, corresponding to points in the parameter plane lying within the ulc, which are attracted to the stationary state from those outside the ulc, which are attracted on to the stable limit cycle and hence which lead to oscillations.

We will frequently wish to determine the amplitude and period of motion around a limit cycle, corresponding to oscillatory behaviour. The simplest approach is just to integrate the governing equations for long enough, with time running forwards if we want to look at stable limit cycles and backwards if we have an unstable cycle. This is not always particularly efficient, and for systems with more than two variables it cannot always be used to obtain... 

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.)...

Even for the present simple case, for which the inflow does not contain the autocatalyst, we have seen a variety of combinations of stable and unstable stationary states with or without stable and unstable limit cycles. Stable limit cycles offer the possibility of sustained oscillatory behaviour (and because we are in an open system, these can be sustained indefinitely). A useful way of cataloguing the different possible combinations is to represent the different possible qualitative forms for the phase plane . The phase plane for this model is a two-dimensional surface of a plotted against j8. As these concentrations vary in time, they also vary with respect to each other. The projection of this motion onto the a-/ plane then draws out a trajectory . Stationary states are represented as points, to which or from which the trajectories tend. If the system has only one stationary state for a given combination of k2 and Tres, there is only one such stationary point. (For the present model the only unique state is the no conversion solution this would have the coordinates a,s = 1, Pss = 0.) If the values of k2 and tres are such that the system is lying at some point along an isola, there will be three stationary states on the phase... 

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. 

FlG. 8.14. The different phase plane portraits identified for cubic autocatalysis with decay (a) unique stable state (b) unique unstable stationary state with stable limit cycle (c) unique stable state with unstable and stable limit cycles (d) two stable stationary states and saddle point (e) stable and unstable states with saddle point (f) stable state, saddle point, and unstable state surrounded by stable limit cycle (g) two unstable states and a saddle point, all surrounded by stable limit cylcle (h) two stable states, one surrounded by an unstable limit cycle, and a saddle point (i) stable state surrounded by unstable limit cycle, unstable state, and saddle point, all surrounded by stable limit cycle (j) stable state, unstable state, and saddle point, all surrounded by stable limit cycle (k) stable state, saddle point, and unstable state, the latter surrounded by concentric stable and unstable limit cycles (1) two stable states, one surrounded by concentric unstable and stable limit cycles, and a saddle point. 

Fig. 12.4. Stationary-state solutions and limit cycles for surface reaction model in presence of catalyst poison K3 = 9, k2 = 1, k3 = 0.018. There is a Hopf bifurcation on the lowest branch p = 0.0237. The resulting stable limit cycle grows as the dimensionless partial pressure increases and forms a homoclinic orbit when p = 0.0247 (see inset). The saddle-node bifurcation point is at...

With p = 0.019, the traverse cuts both Hopf curves, so the stationary-state locus has four Hopf bifurcation points, as shown in Fig. 12.6(c), each one supercritical. There are two separate ranges of the partial pressure of R over which a stable limit cycle and hence sustained oscillations occur. 

Fig. 13.3. (a), (b) Phase plane and quasi-three-dimensional representation of a stable singular point for a two-dimensional system, (c), (d) Equivalent forms for a stable limit cycle surrounding... 

We now have a total of six parameters four from the autonomous system (p, r0, and the desorption rate constants k, and k2) and two from the forcing (rf and co). The main point of interest here is the influence of the imposed forcing on the natural oscillations. Thus, we will take just one set of the autonomous parameters and then vary rf and co. Specifically, we take p = 0.019, r0 = 0.028, fq = 0.001, and k2 = 0.002. For these values the unforced model has a unique unstable stationary state surrounded by a stable limit cycle. The natural oscillation of the system has a period t0 = 911.98, corresponding to a natural frequency of co0 = 0.006 889 6. 

Let us imagine a scenario for which a supercritical Hopf bifurcation occurs as one of the parameters, fi say, is increased. For fi < fi, the stationary state is locally stable. At fi there is a Hopf bifurcation the stationary state loses stability and a stable limit cycle emerges. The limit cycle grows as ft increases above fi. It is quite possible for there to be further bifurcations in the system if we continue to vary fi. With three variables we might expect to have period-doubling sequences or transitions to quasi-periodicity such as those seen with the forced oscillator of the previous section. Such bifurcations, however, will not be signified by any change in the local stability of the stationary state. These are bifurcations from the oscillatory solution, and so we must test the local stability of the limit cycle. We now consider how to do this. 

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