Oscillations, limit cycle

Molecular models for circadian rhythms were initially proposed [107] for circadian oscillations of the PER protein and its mRNA in Drosophila, the first organism for which detailed information on the oscillatory mechanism became available [100]. The case of circadian rhythms in Drosophila illustrates how the need to incorporate experimental advances leads to a progressive increase in the complexity of theoretical models. A first model governed by a set of five kinetic equations is shown in Fig. 3A it is based on the negative control exerted by the PER protein on the expression of the per gene [107]. Numerical simulations show that for appropriate parameter values, the steady state becomes unstable and limit cycle oscillations appear (Fig. 1). 

If the cell cycle in amphibian embryonic cells appears to be driven by a limit cycle oscillator, the question arises as to the precise dynamical nature of more complex cell cycles in yeast and somatic cells. Novak et al. [144] constructed a detailed bifurcation diagram for the yeast cell cycle, piecing together the diagrams obtained as a function of increasing cell mass for the transitions between the successive phases of the cell cycle. In these studies, cell mass plays the role of control parameter a critical mass has to be reached for cell division to occur, provided that it coincides with a surge in cdkl activity which triggers the G2/M transition. 

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 13.10 shows a representation of the phase plane behaviour appropriate to small-amplitude forcing. There are two basic cycles which make up the full motion first, there is the natural limit cycle, corresponding for example to Fig. 13.9(a) around which the unforced system moves secondly, there is a small cycle, perpendicular to the limit cycle, corresponding to the periodic forcing term. The overall motion, obtained as the small cycle is swept around the large one, gives a torus and the buckled limit cycle oscillations at low rf in Fig. 13.9 draw out a path over the surface of such a torus. 

FIGURE 2 The birth and growth of limit cycle oscillations in the I - a, jS, Tr space for a system with non-zero e and k displaying a mushroom stationary-state pattern. Oscillatory behaviour originates from a supercritical Hopf bifurcation along the upper branch and terminates via homoclinic orbit formation. 

Taylor, T. W. and Geiseler, W., 1986, Periodic operation of a stirred flow reaction with limit cycle oscillations. Ber. Bunsenges Phys. Chem. (submitted). 

With A = 0.06 M and the rate constants of Ref. 14b, these equations admit a unique homogeneous steady-state solution (HSS). It is well known that the irreversible Oregonator 14 and its reversible counterpartl4b exhibit homogeneous limit cycle oscillations for realistic values of rate constants and buffered concentrations. My purpose here is to explore several other features of the reversible model (F) which explain a variety of observed behaviors in closed and open stirred reactors. To that end I begin with the stability properties of the unique HSS, as displayed in the partial phase diagram of Fig. 1. 

The Van der Pol oscillator is a well-known and best studied example of a limit cycle oscillation. It has its origin in nonlinear electric circuits. We take a generalized version of it, which in its normalized form reads ... 

Figure 5. v(t) — t (amplitude as a junction of time) and v(t) — 8(%) (phase-plane) diagrams of the coherent oscillation model (Model 2) with F0 == 0 = A. The computer plot shows a typical limit cycle oscillation. 

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.

Figure 12.6a shows the temporal variation of the proximal tubular pressure Pt as obtained from the single-nephron model for a = 12 and T = 16 s. All other parameters attain their standard values as listed in Table 12.1. Under these conditions the system operates slightly beyond the Hopf bifurcation point, and the depicted pressure variations represent the steady-state limit cycle oscillations reached after the initial transient has died out For physiologically realistic parameter values the model reproduces the observed self-sustained oscillations with characteristic periods of 30-40 s. The amplitudes in the pressure variation also correspond to experimentally observed values. Figure 12.6b shows the phase plot Here, we have displayed the normalized arteriolar radius r against the proximal intratubular pressure. Again, the amplitude in the variations of r appears reasonable. The motion... 

Consequently, the amplitude of a linear oscillation is set entirely by its initial conditions any slight disturbance to the amplitude will persist forever. In contrast, limit cycle oscillations are determined by the structure of the system itself. 

In the BZ reaction, malonic acid is oxidized in an acidic medium by bromate ions, with or without a catalyst (usually cerous or ferrous ions). It has been known since the 1950s that this reaction can exhibit limit-cycle oscillations, as discussed in Section 8,3. By the 1970s, it became natural to inquire whether the BZ reaction could also become chaotic under appropriate conditions. Chemical chaos was first reported by Schmitz, Graziani, and Hudson (1977), but their results left room for skepticism—some chemists suspected that the observed complex dynamics might be due instead to uncontrolled fluctuations in experimental control parameters. What was needed was some demonstration that the dynamics obeyed the newly emerging laws of chaos. 

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... 

Limit cycle oscillations obtained as a solution of this system are shown as projections of the x,y,z-phase space on the x,y and y,z phase planes, Fig. III.8a and HI.8b, respectively. 

A simulation result exhibiting limit cycle oscillations is given in Fig. III.15. 

Fig. ffl.15. Limit cycle oscillations on x,y-plane. (After Yang (1974))... 

In their analysis they observed limit cycle oscillations. 

Fig. III. 17. Limit cycle oscillations resulting from the effect of the reactor volume V on the stability of steady states. (After Hlavacek and Votruba (1978))...

Using an opposite approach, that of starting from a mathematical solution and designing an experiment, Olsen and Degn (1972) showed that abstract models may lead to an understanding of the oscillatory chemical reactions exhibiting not only just limit cycle oscillations but also chaotic attractor-type oscillations. 

III B) 1980 Boissonade, J., De Kepper, P. Transitions from Bistability to Limit Cycle Oscillations. Theoretical Analysis and Experimental Evidence in an Open Chemical System. J. Phys. Chem. vol. 84, 501-506... 

Ivanov et al. (1980) modelled a class of Langmuir-Hinshelwood reactions, and by analyzing the mathematical model given in two dimensions, they obtained limit cycle oscillations and showed that the influence of adsorbed species on the catalytic reaction rate may lead to periodic oscillations. 

R.E. Mirollo and S.H. Strogatz. Amplitude death in an array of limit-cycle oscillators. J. Stat. Phys., 60 245-262, 1990. 

D. V. Ramana Reddy, A. Sen, and G. L. Johnston. Time delay induced death in coupled limit cycle oscillators. Phys. Rev. Lett., 80 5109-5112, 1998. 

The influence of noise on a dynamical system may have two counteracting effects. On the one hand if the underlying deterministic systems is already oscillatory, like a limit cycle oscillator or a chaotic oscillator, one expects these oscillations to become less regular due to the influence of the noise. On the other hand oscillatory behavior can also be generated by the noise in systems which deterministically do not show any oscillations. A prominent example are excitable systems but also the noise induced hopping between the attractors in a bistable system can be considered as oscillations [1]. 

Let us consider a stochastic system, for which it is possible to define a cycle, i.e. some behavior which repeatedly happens. This can be for example one turn of a limit cycle oscillator, the hopping from one attractor to the other and back again in a bistable system or an excitation from the rest state to the excited state, followed by a relaxation back to the rest state in an excitable system. As in the case of deterministic systems, forced synchronization of stochastic systems is also considered as an adaption of the cyclic motion to the periodic driving. However due to the stochasticity one can never expect perfect synchronization. Instead there is always a finite probability that an additional or missing cycle of the system with respect to the signal happens. The rarer these phase slips occur, the better is the synchronization. Thus synchronization in periodically driven stochastic systems is not an all or none notion but gradually varies from no synchronization to synchronization. 

For a mathematical treatment of synchronization we recall that the phase of an oscillator is neutrally stable and can be adjusted by a small action, whereas the amplitude is stable. This property allows a description of the effect of small forcing/coupling within the framework of the phase approximation. Considering the simplest case of a limit cycle oscillator, driven by a periodic force with frequency u> and amplitude e, we can write the equation for the perturbed phase dynamics in the form... 

The major new property, when including a type-III food uptake in the predator prey model (15.5) is that the prey isocline becomes a cubic-like function. Again, the dynamics depend on the exact location of the intersections of predator- and prey-isoclines. If the intersection is in one of the two decreasing branches of the prey isocline (see Fig. 15.7), the model exhibits a stable fixed point. Otherwise, the fixed point becomes unstable giving rise to limit cycle oscillations (see Fig. 15.6). 

The molecular bases of Ca oscillations, as well as those of the mitotic oscillator, are addressed at the end of this book, where minimal models closely related to recent experimental observations are analysed for the two phenomena and shown to admit limit cycle oscillations. 

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