Limit cycles existence

It describes an open system with a net transport from A to E and from B to D. The model is somewhat unrealistic inasmuch as it involves a reaction for which three molecules must collide. Another model, not having this drawback, is the Oregonator n), which, however, has the added complication that three compounds X, Y, Z are involved. This is inevitable, for it can be proved that no limit cycles exist in systems involving just two reactants with only bimolecular reactions. ... 

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... 

In Figs. 30 and 31, the flow under Eq. (102) is displayed in the phase space (x,y). Figure 30 shows a case where the reaction terminates at the hxed point, and Fig. 31 shows a case where the limit cycle exists. In these hgures, the NHlMs Mo are graphs x = x(y) that are obtained by solving /(x, y) = 0 for x. 

Figure 31. Flow of BZ reaction when limit cycle exists.

The limit cycle found in the previous section holds only for 103 — 031 small. Obviously, once the limit cycle exists, it can be continued, either globally or until certain bad things happen such as the period tending to infinity or the orbit collapsing to a point. It is very difficult to show analytically that these events do not occur. Moreover, the computations necessary to actually prove the asymptotic stability of the bifurcating orbit are very difficult. We discuss briefly some numerical computations, shown in Figure 8.1, which suggest answers to both these problems. 

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. 

Experiment 3. Sustained cycles occur for a parameter combination that satisfies the assumptions of the limit cycle existence theorem. Figure 5.6a exhibits the limit cycle (see also Fig. 5.6 b). Again, the origin is the only singular point which, however, has now become an unstable focus of the motion of the economy. 

Here, /xi = 0 corresponds to the curve Ci the curve C21 also leaves the point (0,0) towards negative values of H2 the curves C21 and L intersect at infinitely many points. Then, one may continue inductively according to this rule let (p, s) be an admissible pair of words ended by 1 then Cp and Cs intersect at infinitely many points, and if P and Q are neighboring points of intersection of Cp and Cs such that hp > hs on the interval between P and Q, then P and Q are further connected by arcs of Cps and Csp which intersect Cs and, respectively, Cp infinitely many times, etc. In the region bounded by these arcs of Cps and a stable limit cycle exists with the code ps. On the contrary, if hp < hs on the interval between P and Q, then for these values... 

A stable limit cycle (Fig. 9.25) is a mathematical model for self-sustained oscillations. According to Poincare and Andronov, a limit cycle exists when... 

Remark 6.6. The preceding analysis shows that there exists a stabilizing damping level, Cst, such that the system equilibrium point is stable for c > Cst and no limit cycles exist. Cjt = 0 in Case 1 and Cases 3 and 4/Scenario 2. Cjt = in Case 2 and Cases 3 and 4/Scenario 3, where Ca is given by (6.48). Cjt > 0 in Cases 3 and 4/Scenario 1. ... 

Hopf bifurcation of the original system [69]. The unstable branch, shown by the dotted line, determines the domain of attraction of the trivial or steady-shding equilibrium point. As discussed by Remark 6.6, for c > Cst the steady-shding state is stable and no limit cycles exist. 

The simplest possible attraetor is a fixed point, for which all trajectories starting from the appropriate basin-of-attraction eventually converge onto a single point. For linear dissipative dynamical systems, fixed-point attractors are in fact the only possible type of attractor. Non-linear systems, on the other hand, harbor a much richer spectrum of attractor-types. For example, in addition to fixed-points, there may exist periodic attractors such as limit cycles for two-dimensional flows or doubly periodic orbits for three-dimensional flows. There is also an intriguing class of attractors that have a very complicated geometric structure called strange attractors [ruelleSO],... 

The spatial and temporal dimensions provide a convenient quantitative characterization of the various classes of large time behavior. The homogeneous final states of class cl CA, for example, are characterized by d l = dll = dmeas = dmeas = 0 such states are obviously analogous to limit point attractors in continuous systems. Similarly, the periodic final states of class c2 CA are analogous to limit cycles, although there does not typically exist a unique invariant probability measure on... 

Let P a a ) be the probability of transition from state a to state a. In general, the set of transition probabilities will define a system that is not describ-able by an equilibrium statistical mechanics. Instead, it might give rise to limit cycles or even chaotic behavior. Fortunately, there exists a simple condition called detailed balance such that, if satisfied, guarantees that the evolution will lead to the desired thermal equilibrium. Detailed balance requires that the average number of transitions from a to a equal the number of transitions from a to a ... 

What is really difficult is to ascertain whether a given differential equation has a limit cycle solution. In fact, aside from a few well-known equations (of van der Pol, Rayleigh, Li6nard, etc.) we hardly know anything at all about the existence of cycles directly, that is, on the basis of the topological methods with which we are concerned here. 

The Index of Poincard.—The concept of the index was introduced by Poincar6 for the purpose of establishing a necessary criterion for the existence of a closed trajectory (the limit cycle). 

The dynamic behavior of nonisothermal CSTRs is extremely complex and has received considerable academic study. Systems exist that have only a metastable state and no stable steady states. Included in this class are some chemical oscillators that operate in a reproducible limit cycle about their metastable... 

Cutlip and Kenney (44) have observed isothermal limit cycles in the oxidation of CO over 0.5% Pt/Al203 in a gradientless reactor only in the presence of added 1-butene. Without butene there were no oscillations although regions of multiple steady states exist. Dwyer (22) has followed the surface CO infrared adsorption band and found that it was in phase with the gas-phase concentration. Kurtanjek et al. (45) have studied hydrogen oxidation over Ni and have also taken the logical step of following the surface concentration. Contact potential difference was used to follow the oxidation state of the nickel surface. Under some conditions, oscillations were observed on the surface when none were detected in the gas phase. Recently, Sheintuch (46) has made additional studies of CO oxidation over Pt foil. 

When the steady state becomes unstable, the system moves away from it and often undergoes sustained oscillations around the unstable steady state. In the phase space defined by the system s variables, sustained oscillations generally correspond to the evolution toward a limit cycle (Fig. 1). Evolution toward a limit cycle is not the only possible behavior when a steady state becomes unstable in a spatially homogeneous system. The system may evolve toward another stable steady state— when such a state exists. The most common case of multiple steady states, referred to as bistability, is of two stable steady states separated by an unstable one. This phenomenon is thought to play a role in differentiation [30]. When spatial inhomogeneities develop, instabilities may lead to the emergence of spatial or spatiotemporal dissipative stmctures [15]. These can take the form of propagating concentration waves, which are closely related to oscillations. 

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]. 

The periodic recurrence of cell division suggests that globally the cell cycle functions like an autonomous oscillator. An extended model incorporating the sequential activation of the various cyclin-dependent kinases, followed by their inactivation, shows that even in the absence of control by cell mass, this sequence of biochemical events can operate as a limit cycle oscillator [145]. This supports the union of the two views of the cell cycle as dominoes and clock [146]. Because of the existence of checkpoints, however, the cell cycle stops at the end of certain phases before engaging in the next one. Thus the cell cycle looks more like an oscillator that slows down and makes occasional stops. A metaphor for such behavior is provided by the movement of the round plate on the table in a Chinese restaurant, which would rotate continuously under the movement imparted by the participants, were it not for frequent stops. 

I would like to comment on the theoretical analysis of two systems described by Professor Hess, in order to relate the phenomena discussed by Professor Prigogine to the nonequilibrium behavior of biochemical systems. The mechanism of instability in glycolysis is relatively simple, as it involves a limited number of variables. An allosteric model for the phosphofrucktokinase reaction (PFK) has been analyzed, based on the activation of the enzyme by a reaction product. There exists a parameter domain in which the stationary state of the system is unstable in these conditions, sustained oscillations of the limit cycle type arise. Theoretical... 

This implies that for i < ic there are no limit cycles in the phase plane u,p). Note that this conclusion relies on (6.2.10) and thus is only true for a symmetric f(u). For a non-symmetric f(u) by the above argument there are no limit cycles for i different from zero. On the other hand, some limit cycles in this case still might exist in the current range ix < i < ic. 

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. 

Since, as we saw previously, for a general nonsymmetric f(u) (/"(0) / 0) the limit cycles may exist already in the current range... 

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... 

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. 

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). 

Exercise. Take in (7.3) the special values a = 1, ft = 3. Prove in the following way that there is a limit cycle. There is one unstable stationary solution. On the other hand, there exists a closed curve surrounding it with the property that all solutions it intersects are directed towards its interior viz., the curve formed by the axes and the lines y = 5.84 + x, y = 7.84 — x. 

In the case of a chemical clock, the asymptotic (f -A oo) solution depends on time, there are not only singular points but also singular trajectories. An example is the stable limit cycle - Fig. 2.4, i.e., a closed trajectory to which all phase trajectories existing in its vicinity strive. 

Note here, without proof, one of the synergetic theorems about limit cycles [14, 15] a stable limit cycle contains at least one singular point or the unstable node of focus-type exists. 

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