Oscillations stable limit cycle

The prototype, cubic autocatalytic reaction (A + 2B 3B) forms the basis of a simple homogeneous system displaying a rich variety of complex behaviour. Even under well-stirred, isothermal open conditions (the CSTR) we may find multi stability, hysteresis, extinction and ignition. Allowing for the finite lifetime of the catalyst (B inert products) adds another dimension. The dependence of the stationary-states on residence-time now yields isolas and mushrooms. Sustained oscillations (stable limit cycles) are also possible. There are strong analogies between this simple system and the exothermic, first-order reaction in a CSTR. 

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

If the roots are pure imaginary numbers, the form of the response is purely oscillatory, and the magnitude will neither increase nor decay. The response, thus, remains in the neighbourhood of the steady-state solution and forms stable oscillations or limit cycles. 

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]. 

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. ...

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. 

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.

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. 

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. 

Furthermore, the oscillation starting with even larger amplitudes gets the kinetic energies until it reaches a limit where the oscillation induces strong shock waves and dissipates its kinetic energies (see Fig. 1). Thus we conclude that the model which has the stable limit cycle near the transition has another unstable fixed point with a larger amplitude. The transition therefore is induced by the... 

Let us now formulate the problem of the energy-optimal steering of the motion from a chaotic attractor to the coexisting stable limit cycle for a simple model, a noncentrosymmetric Duffing oscillator. This is the model that, in the absence of fluctuations, has traditionally been considered in connection with a variety of problems in nonlinear optics [166]. Consider the motion of a periodically driven nonlinear oscillator under control... 

Closed trajectories around the whirl-type non-rough points cannot be mathematical models for sustained self-oscillations since there exists a wide range over which neither amplitude nor self-oscillation period depends on both initial conditions and system parameters. According to Andronov et al., the stable limit cycles are a mathematical model for self-oscillations. These are isolated closed-phase trajectories with inner and outer sides approached by spiral-shape phase trajectories. The literature lacks general approaches to finding limit cycles. 

For the oscillations we have discussed so far, the only requirement on the interfacial kinetics of the system is that it possess an N-NDR. Oscillations come into play as a result of the interplay of the interfacial kinetics with ohmic losses and transport limitations. Hence, for nearly every electron-transfer system that possesses an N-NDR, conditions can be set up under which stable limit cycles exist, and many experimentally observed oscillations could be traced back to this mechanism. Overviews of these experimental systems can be found in Refs. [9, 10, 68], Here we compile only a few examples. Figure 13 shows experimental cyclic voltammograms of the reduction... 

The form of the solutions to the simplified model were analysed by examining the existence and types of the pseudo-stationary points of the equations for d0/dr = d 3/dr = 0 and values of e in the range 0—1 (r = Figure 29 shows the oscillation of a multiple-cool-flame solution about the locus of such a pseudo-stationary point, Sj. The initial oscillation is damped while Si is a stable focus. The changing of Si into a unstable focus surrounded by a stable limit cycle leads to an amplification of the oscillation which approaches the amplitude of the limit cycle. When Si reverts to a stable focus, and then a stable node, the solution approaches the locus of the pseudo-stationary point. In this way an insight may be gained into the oscillatory behaviour of multiple cool flames. 

The previous two chapters showed that competitive exclusion holds under a variety of conditions in the chemostat and its modifications. In this chapter it will be shown that if the competition is moved up one level - if the competition occurs among predators of an organism growing on the nutrient - then coexistence may occur. The fact that the competitors are at a higher trophic level allows for oscillations, and the coexistence that occurs is in the form of a stable limit cycle. Along the way it will be necessary to study a three-level food-chain problem which is of interest in its own right it is the chemostat version of predator-prey equations. The presentation follows that of [BHWl]. 

Fig. 5.9. Variation of (a) concentration y and (b) temperature excess 6 in time for a value of in region of steady-state instability showing sustained oscillations (c) typical limit cycle lying around unstable steady-state o produced by plotting y against (d) for some parameter values there are two limit cycles surrounding a stable steady-state, one unstable (broken curve) the other stable (solid curve). In (c) and (d) example trajectories are shown (thin curves) that wind either onto the stable limit cycle or, in (d) onto the stable steady-state...

Stable, limit cycle. The latter occurs in the Salnikov case and the modified bifurcation diagram is shown in Fig. 5.11(b). The stable limit cycle born at the lower Hopf point overshoots the upper Hopf point but is extinguished by colliding with the unstable limit cycle born at the upper Hopf point which also grows in amplitude as )jl is increased. Over a, typically narrow range, then there are two limit cycles, one unstable and one stable around the (stable) steady-state point. If we start with the system at some large value of /r, so we settle onto the steady-state locus, and then decrease the parameter, we will first swap to oscillations at the Hopf point /r - At this point there is a stable limit cycle available as the system departs from the now unstable steady-state, but this stable limit cycle is not born at this point and so already has a relatively large amplitude. We would expect to... 

As you might guess, the system eventually settles into a self-sustained oscillation where the energy dissipated over one cycle balances the energy pumped in. This idea can be made rigorous, and with quite a bit of work, one can prove that the van der Pol equation has a unique, stable limit cycle for each fj >0. This result follows from a more general theorem discussed in Section 7.4. 

Use two-timing to show that the van der Pol oscillator (2)has a stable limit cycle... 

Figure 2 shows a time trace of the dynamics for N = 3, n = 6. Now the dynamics follow a stable limit cycle oscillation. This forms the basis for the synthesis of the repressilator [27]. In this case the eigenvalues of the hxed point atx] = X2 = X3 = 1/2 are = —1 — n/2 and X23 = —1 + n/4 f3nijA [34]. In this case there is a Hopf bifurcation when n = 4 so that for values of n > 4 there is a stable limit cycle oscillation corresponding to the repressilator. If the equations for the dynamics of mRNA are included, then oscillations are still found, but now oscillations can be found for smaller values of the Hill... 

It was pointed out earlier that oscillations in NDR oscillators are linked to three features of the electrochemical system (1) an N-shaped steady-state polarization curve (2) a resistance in series with the working electrode, which must not be too large and (3) a slow recovery of the electroactive species, in most cases due to slow mass transport. Hence, for every system that was discussed in the context of the possible origin of N-shaped characteristics, conditions can be estabhshed under which stable limit cycles exist, and for most of the systems mentioned, oscillations were in fact observed. This unifying approach was first put forth by Koper and Sluyters, and numerous experimental examples of electrochemical oscillations that can be deduced according to this mechanism are discussed in Ref. 60. 

This model was slightly altered by Sel kov and Betz (1973). The model still has only one singular solution, and a stable limit cycle corresponding to oscillations. 

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