Limit cycle oscillation existence

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. 

Figure 12.2b shows such a power spectrum for the tubular pressure variations depicted in Fig. 12.2a. This spectrum demonstrates the existence of two main and clearly separated peaks a slow oscillation with a frequency fsiow 0.034 Hz that we identify with the TGF-mediated oscillations, and a significantly faster component at ffasl 0.16 Hz representing the myogenic oscillations of the afferent arteriole. Both components play an essential role in the description of the physiological control system. The power spectrum also shows a number of minor peaks on either side of the TGF peak. Some of these peaks may be harmonics (/ 0.07 Hz) and subharmonics (/ 0.017 Hz) of the TGF peak, illustrating the nonlinear character of the limit cycle oscillations.

Working with the model of B-Z reaction, Sakanoue and Endo (1982) showed by computer simulation the coexistence of a stable and an unstable limit cycle. The existence of an unstable oscillating object between two stable objects had been... 

Since 5>0, d[fip),f(q)]contraction mapping (see Appendix) and there must be a unique stable limit cycle in the four regions. In Fig. 14, we give this construction for the parameters used to compute Fig. 12. Although there is good agreement between the dynamics in the piecewise linear and the continuous equations, no proof of stable limit cycle oscillations has been found for Eq. (48) or (50). However, there has been a recent proof for the existence of nonlocal periodic solutions of Eq. (45) using fixed-point methods. ... 

Whichever qualitative features are examined, the apparently rich chemical literature reduces to a few simple classes. It is difficult to judge the reason why so few dynamic classes have appeared so far. Perhaps there are certain (still unanalyzed) features of kinetic equations that lead to simple dynamics. Another possibility is that chemists have tended to study only a small subclass of chemical kinetics and have ignored the rest. For example, a type of dynamical behavior that it is hard to imagine not existing is one in which there are two stable attractors. A limit cycle oscillation and a stable steady state where transitions can occur between the two as a result of large perturbations of concentrations. An example has been previously given in which this type of... 

The population models of limit cycle oscillators which we obtained in Sect. 5.2 (c) seem to have been seldom investigated in the past. Although a general analytical treatment of (5.2.21) would be difficult, there certainly exists, in the limit of large N, a special subclass of systems for which a number of interesting analytical results are available. 

We first consider an apparently simple situation a trigger wave in an oscillatory medium. The period-1 limit cycle that exists for small K2 values to the left of the period doubling cascade shown in Figure 2 takes on a relaxation character as 2 is increased towards period-2. If the temporal profile of such an oscillation is transcribed into the spatial domain a trigger wave is formed as is shown in the top panel of Figure 7. To obtain the results in this panel... 

Further stochastic simulation studies now in progress are concerned with fluctuation and nucleation in evolving chemical systems (e.g., limit cycle oscillations, combustion and explosions) and at the transition to spatial dissipative structure (cf.. Figure 10). In the latter case, for example, stochastic simulations verify the existence of critical long-range spatial correlations predicted in a stochastic theoretical study of the model (cf.. Ref. 17). 

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]. 

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... 

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 second important bifurcation that is connected with a stability change in a stationary state is the /fop/bifurcation. At a Hopf bifurcation, the real parts of two conjugate complex eigenvalues of J vanish, and as Hopf s theorem ensures, a periodic orbit or limit cycle is bom. A limit cycle is a closed loop in phase space toward which neighboring points (of the kinetic representation) are attracted or from which they are repelled. If all neighboring points are attracted to the limit cycle, it is stable otherwise it is unstable (see Ref. 57). The periodic orbit emerging from a Hopf bifurcation can be stable or unstable and the existence of a Hopf bifurcation cannot be deduced from the mere fact that a system exhibits oscillatory behavior. Still, in a system with a sufficient number of parameters, the presence or absence of a Hopf bifurcation is indicative of the presence or absence of stable oscillations. 

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. 

Salnikov specifically reported multiple singular points and a limit cycle establishing the existence of oscillations in chemical reactions. Bilous and Amundson (1955) referred to Salnikov s (1948) paper as the first work where periodic phenomenon in reaction systems was discussed. They also indicated that a reaction A -> B in CSTR is irreversible, exothermic, and kinetically first order. Considering mass balance and heat balance equations it is known that at the steady states, the heat consumption... 

To study a class of mechanisms for isothermal heterogeneous catalysis in a CSTR, Morton and Goodman (1981-1) analyzed the stability and bifurcation of simple models. The limit cycle solutions of the governing mass balance equations were shown to exist. An elementary step model with the stoichiometry of CO oxidation was shown to exhibit oscillations at suitable parameter values. By computer simulation limit cycles were obtained. 

The solutions of (3.68)-(3.69) for positive parameters and generic initial conditions are oscillations, of amplitude fixed by the initial conditions, around the fixed point Z — a /a2, P = b2/b. The equation of this family of closed trajectories is ai In Z + b2 In P — biP — a2Z = constant. The oscillations are suggestive of the population oscillations observed in some real predator-prey systems, but they suffer from an important drawback the existence of the continuous family of oscillating trajectories is structurally unstable systems similar to (3.68)-(3.69) but with small additional terms either lack the oscillations, or a single limit cycle is selected out of the continuum. Thus, the model (3.68)-(3.69) can not be considered a robust model of biological interactions, which are never known with enough... 

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