Oscillation limit cycle type

The deep philosophical significance of the new theory lies precisely at this point, and consists in replacing a somewhat metaphysical concept of the harmonic oscillator (which could never be produced experimentally) by the new concept of a physical oscillator of the limit cycle type, with which we are dealing in the form of electron tube circuits and similar self-excited systems. 

Summing up, everything which oscillates in a stationary state in the world around us is necessarily of the limit cycle type it depends only on the parameters of the system, that is, on the differential equation, and not on the initial conditions. 

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. 

At each point in space the oscillations are similar to the limit cycle type of behavior in ordinary differential equations. The various regions of the... 

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... 

Nonlinear Oscillations (Limit Cycles). We want to restrict ourselves to nonlinear oscillations of limit cycle type (LC), which means that we are only dealing with selfsustained oscillations. This type of nonlinear oscillations can only occur in nonconservative systems, it is a periodic process, which is produced at the expense of a nonperiodic source of energy within the system. 

Fig. 6.19. Evolution in the phase space at different values of the relative proportions of (initially) chaotic and periodic cell populations in a mixed suspension containing various amounts of the two types of cells, (a) Oscillations of the limit cycle type obtained for Vj = 4.5 x 10 min" when the suspension contains only cells of periodic population 2 (fj = 0, fj = l)l arrows show the direction of movement and the trajectory has been broken to indicate the part that comes behind (the portion of the curve in front corresponds to a decrease in all three variables after a peak in cAMP). (b) Period-2 oscillations obtained upon adding to periodic population 2 cells from the chaotic population 1, for which Vi = 4.396875 x 10" min" the value of the fraction of the (initially), chaotic population is = 0.5. (c) Period-4 oscillations obtained when is increased up to 0.86 notice that two of the loops of the trajectory over a period are very close to each other, which is also apparent in the bifurcation diagram of fig. 6.20. (d) Chaotic behaviour corresponding to a strange attractor when the suspension contains only cells of population 1 (Fj = 1). The curves are obtained by numerical integration of eqns (6.9) for the above-indicated values of Vj and Vj other parameter values, which hold for the two populations, are as in fig. 6.2. Variables pr and a relate to population 2 in (a)-(c), and to the homogeneous population 1 in (d) variable y is shared by the two populations. Ranges of variation for pr, a and y are 0-1,0.65-0.68 and 0-2.2, respectively. Initial conditions were a = 0.6729 and pr = 0.2446 for both populations, while 7=1.7033. The curves were obtained after a transient of 500-1000 min. The period of the oscillations shown in (a)-(c) is of the order of 8-10 min thus for F. = 0.3 and Fj = 0.7 the period is equal to 8.7 min (Halloy et al. 1990).

The study of the stability properties of the unique steady state admitted by eqns (5.1) permits us to establish the stability diagram of fig. 7.2 in which the dashed area C denotes the domain of instability of the steady state, where sustained oscillations of the limit cycle type occur. In domain B, the steady state is stable but excitable, as the system amplifies, in a pulsatory manner, a cAMP signal whose given amplitude exceeds a threshold. Everywhere else in the diagram the steady state is stable but nonexcitable. 

The hypothesis that a biochemical oscillator of the limit cycle type controls the onset of mitosis has been proposed for long (Sel kov, 1970 Gilbert, 1974,1978 Winfree, 1980,1984 for an earlier discussion of cell division in terms of a chemical oscillatory process, see Rashevsky, 1948) and was put forward, in particular detail, on the basis of experiments performed on the slime mould Physarum (Kauffman, 1974 Kauffman... 

The use of models based on experimental observations has shown how the regulatory properties at the level of enzymes and receptors can give rise to nonequilibrium, temporal self-organization in the form of sustained oscillations of the limit cycle type. In this, sustained oscillations represent examples of the dissipative structures described by Prigogine (1969). Like spatial or spatiotemporal dissipative structures, limit cycle oscillations occur beyond a critical point of instability and... 

The examples of rhythmic behaviour analysed in this book all belong to dynamics of the limit cycle type. In the case of phosphofructokinase, like in that of cAMP synthesis in Dictyostelium or signal-induced Ca oscillations, the analysis of models based on experimental data indeed shows that these systems admit a nonequilibrium steady state that becomes unstable beyond a critical value of some control parameter. It is in these conditions that sustained oscillations occur, in the form of a limit cycle in the phase space. 

Thus we see how certain biological rhythms can be viewed as a cyclic succession of discontinuously linked events rather than as continuous oscillations of the limit cycle type. As demonstrated by the biochemical... 

Fig. 12.1. Biochemical models based on various modes of positive feedback in enzyme reactions. All these models admit simple periodic behaviour of the limit cycle type. The coexistence between two stable limit cycles (birhythmidty) or between a stable limit cycle and a stable steady state (hard excitation) is observed in models (b) to (d). Model (d) eilso admits complex periodic oscillations of the bursting type, chaos, as well as the coexistence between three simultaneously stable limit cycles (trirhythmicity) (Goldbeter et oL, 1988).

The main purpose of this paper is to present a simple theoretical model to describe this interesting phenomenon which provides a further example of the synergy between an equilibrium phase transition and a pattern forming instability. It is based on a Landau type equation for the polymer volume fraction coupled to a simple two variables reactive system that, on its own, can undergo a Hopf bifurcation giving rise to chemical oscillations of the limit cycle type. These oscillating systems have now been extensively studied both from the theoretical and experimental points of view (7). 

In spite of the link which might be established between the limit cycle type behaviour and the occurrence of target patterns, it is tempting to search very quickly for some qualitative correlation between the two. One can hopefully imagine, for instance, that a difference in the properties of the limit cycle has something to do with the above-mentioned fact that oscillations in thin layer do not necessarily give rise to centres. 

A numerical calculation evidences the non stationary behaviour. Sustained oscillations of the surface values, of the limit—cycle type, propagate in the scale by getting damped. Consequently the reaction rate oscillates and the kinetics of growth of the oxidized... 

Controlled chaos may also factor into the generation of rhythmic behavior in living systems. A recently proposed modeL describes the central circadian oscillator as a chaotic attractor. Limit cycle mechanisms have been previously offered to explain circadian clocks and related phenomena, but they are limited to a single stable periodic behavior. In contrast, a chaotic attractor can generate rich dynamic behavior. Attractive features of such a model include versatility of period selection as well as use of control elements of the type already well known for metabolic circuitry. 

This system in its linear version (i.e., when e = 0) is a dynamical filter. Suppose that the oscillators interact with each other with the interaction parameter a = 0.9. The frequency 00 of the external driving field varies in the range 0 < < 4.2. The other parameters of the system are A 200, coq 1, c 0.1, and = 0.05. The autonomized spectrum of Lyapunov exponents A-4, >,5 versus the frequency to is presented in Fig. 23. In the range 0 < < 0.2 the system does not exhibit chaotic oscillation. Here, the maximal Lyapunov exponent Xi = 0 and the spectrum is of the type 0, —, —, —, (limit cycles). 

The oscillations produced by a frequency generator are of this type and have inspired Van der Pol to construct his classic example of a differential equation having a limit cycle. The best known example in chemistry is the Zhabotinskii reaction. In biology many periodic phenomena are known that can presumably be described in this way. ... 

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. 

The rate oscillations produced by the model are always simple relaxation type oscillations (Fig. 5). The model cannot reproduce the rather complex oscillation waveform which was observed experimentally under many operating conditions (Fig. 1). However the model predicts the correct order of magnitude of the limit cycle frequency and also reproduces most of the experimentally observed features of the oscillations figure 2 compares the experimental results of the limit cycle frequency and amplitude (defined as maximum % deviation from the average rate) with the model predictions. The model correctly predicts a decrease in period and amplitude with increasing space velocity at constant T and gas composition. It also describes semiquantitatively the decrease in period and amplitude with increasing temperature at constant space velocity and composition (Fig. 3). 

It is necessary to emphasize one principal peculiarity of the copolymerization dynamics which arises under the transition from the three-component to the four-component systems. While the attractors of the former systems are only SPs and limit cycles (see Fig. 5), for the latter ones we can also expect the realization of other more complex attractors [202]. Two-dimensional surfaces of torus on which the system accomplishes the complex oscillations (which are superpositions of the two simple oscillations with different periods) ate regarded to be trivial examples of such attractors. Other similar attractors are fitted by the superpositions of few simple oscillations, the number of which is arbitrary. And, finally, the most complicated type of dynamic behavior of the system when m 4 is fitted by chaotic oscillations [16], for which a so-called strange attractor is believed to be a mathematical image [206]. 

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