Limit cycle oscillation nonlinear

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 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.

Under these conditions system (9.1) still admits a unique steady state, but linear stability analysis shows that the latter is always stable (Goldbeter Dupont, 1990) this rules out the occurrence of sustained oscillations around a nonequilibrium unstable steady state. This result holds with previous studies of two-variable systems governed by polynomial kinetics these studies indicated that a nonlinearity higher than quadratic is needed for limit cycle oscillations in such systems (Tyson, 1973 Nicolis Prigogine, 1977). Thus, in system (9.1), it is essential for the development of Ca oscillations that the kinetics of pumping or activation be at least of the Michaelian type. Experimental data in fact indicate that these processes are characterized by positive cooperativity associated with values of the respective Hill coefficients well above unity, thus favouring the occurrence of oscillatory behaviour. 

With regard to the role of nonlinear feedback control, periodic behaviour was illustrated in fig. 11.7 for a repression function characterized by a Hill coefficient of 4 however, a value of 2 or 1 for that cooperativity coefficient can also give rise to sustained oscillations. The cooperativity of repression provides a major source of nonlinearity required for sustained oscillatory behaviour. This is the reason why steep thresholds due to zero-order ultrasensitivity are not required here to generate limit cycle oscillations (see the relatively large values of the reduced Michaelis constants K, used in fig. 11.7), in contrast to the situation encountered for the phosphorylation-dephosphorylation cascade model analysed for the mitotic oscillator. 

The techniques described in this section are useful for studying chemical dynamics in the neighborhood of critical points. The remainder of this paper is devoted to the analysis of the global dynamics of nonlinear kinetic equations. In Section 2 a topological theorem is given, which can be used to place restrictions on the entire set of critical points of a chemical network. In Section 3 it is shown that many chemical networks can be classified on the basis of flows between volumes in concentration space. In Section 4, a number of techniques for establishing limit cycle oscillations in three and more dimensions are described. The topological methods are applied to analysis of compartmental chemical systems in Section 5. The results are discussed in Section 6. In the Appendix the principal mathematical results that have been used in the text are summarized. 

Some Limit Cycle Oscillations in Nonlinear Kinetic Equations... 

The periodic behavior at = nfl is structurally unstable the smallest change in the parameter q leads to a qualitative change in the behavior. By adding a small nonlinear term, for example, —ax t), to the right-hand side of eq. (10.1), one obtains a system that gives structurally stable limit cycle oscillations, as illustrated in Figure 10.2, over a range of parameters q (Epstein, 1990). 

Figure 10.2 Limit cycle oscillations obtained by adding a nonlinear term —ax tf, with a 1, to the right-hand side of eq. (10.1). Here q = 1.7. Initial values Xq = 1 (a), 5 (b). and 0.2 (c) all lead to a periodic oscillation with amplitude 1.87 and period 3.76. (Adapted from Epstein, 1990.)...

One important feature of reaction-diffusion fields, not shared by fluid dynamical systems as another representative class of nonlinear fields, is worth mentioning. This is the fact that the total system can be viewed as an assembly of a large number of identical local systems which are coupled (i.e., diffusion-coupled) to each other. Here the local systems are defined as those obeying the diffusionless part of the equations. Take for instance a chemical solution of some oscillating reaction, the best known of which would be the Belousov-Zhabotinsky reaction (Tyson, 1976). Let a small element of the solution be isolated in some way from the bulk medium. Then, it is clear that in this small part a limit cycle oscillation persists. Thus, the total system may be imagined as forming a diffusion-coupled field of similar limit cycle oscillators. 

Let a limit cycle oscillator be exposed to some weak random forces which may depend on the state variable X. The governing equation is a nonlinear stochastic equation ... 

The school from Brussels have found that for the systems maintained far from thermodynamic equilibrium and when the kinetic laws are suitably nonlinear, the excess entropy production, S P becomes negative for t > t. In this case, the nonequilibrium steady state becomes unstable and sometimes, when the internal structure permits, the system executes limit cycle oscillations [1-3]. 

An empirical (black-box) approach to studying rhythmic behavior is to model the oscillation as limit cycles of a nonlinear oscillator. A classic example of limit-cycle oscillation is the Van der Pol oscillator ... 

The single Kerr anharmonic oscillator has one more interesting feature. It is obvious that for Cj = 0 and y- = 0, the Kerr oscillator becomes a simple linear oscillator that in the case of a resonance 00, = (Do manifests a primitive instability in the phase space the phase point draws an expanding spiral. On adding the Kerr nonlinearity, the linear unstable system becomes highly chaotic. For example, putting A t = 200, (D (Dq 1, i = 0.1 and yj = 0, the spectrum of Lyapunov exponents for the first oscillator is 0.20,0, —0.20 1. However, the system does not remain chaotic if we add a small damping. For example, if yj = 0.05, then the spectrum of Lyapunov exponents has the form 0.00, 0.03, 0.12 1, which indicates a limit cycle. 

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... 

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. 

Externally driven limit cycles can exhibit a great variety of behaviour. Quite generally, one gets a nonlinear superposition of an internal nonlinear oscillation with an external oscillatory perturbation. Details of the resulting behaviour (including sub- and superharmonic oscillations, entrainment, quenching...) depend on both, the frequency and intensity of the applied fields and on the internal nonlinear kinetics of the considered system. 

Furthermore, the concept of externally driven limit cycles can yield an explanation of the extremely narrow frequency ranges of these effects. One can easily show that the nonlinear damping term vanishes, if one averages over one period of oscillation. 

Kaiser, F., Nonlinear Oscillations (Limit Cycles) in Physical and Biological Systems p. 343-389, in "Nonlinear Electro-... 

Using the ability to displace the trajectory of a limit cycle across a fixed boundary (the separatrix) as a measure of sensitivity to an external perturbation, it can therefore be seen that nonlinear oscillating reaction systems are able to respond most sensitively to a range of externally applied frequencies close to their endogenous frequency. 

Instead of using this equation, in the literature, there are few models proposed by which the frequency or Strouhal number of the shedding is fixed. Koch (1985) proposed a resonance model that fixes it for a particular location in the wake by a local linear stability analysis. Upstream of this location, flow is absolutely unstable and downstream, the flow displays convective instability. Nishioka Sato (1973) proposed that the frequency selection is based on maximum spatial growth rate in the wake. The vortex shedding phenomenon starts via a linear instability and the limit cycle-like oscillations result from nonlinear super critical stability of the flow, describ-able by Eqn. (5.3.1). 

Control of industrial polymerization reactors is a challenging task because, in general, control engineers lack rigorous polymerization process knowledge, process model, and rapid online or inline sensors to measure polymer properties. Exothermic polymerization processes often exhibit strongly nonlinear dynamic behaviors (e.g., multiple steady states, autonomous oscillations, limit cycles, parametric sensitivity, and thermal runaway), particularly when continuous stirred tank... 

Another phenomenon which these nonlinear equations can show is that of the limit cycle in which the reactor tends to take up a continual, very nonlinear oscillation. An example of this was found when the control of the unstable state B of Fig. 7.18 was studied. For this case L = 2 and L-r M N = LM — N — —2.25. Addition of the control would thus make L -r Me — N = —2.25 + and LMc — N —2.25 and the... 

Crooke et al. (1980) discussed the fermentation process, the growth of yeast, in a continuously stirred tank fed with glucose, minerals and vitamins. Defining X as the concentration of the cells and S as that of the substrate (nutrient) a model consisting of two nonlinear differential equations of the first order was given and sustained oscillations in X and S were obtained. Specifically the cases of one limit cycle and two limit cycles were illustrated. 

The ranges of temperature and butene feed corresponding to different flow rates, were experimentally recorded and regions of oscillations were determined. For a fixed temperature, (150 °C) and 1 % butene feed, corresponding to increasing flow rates different oscillatory behaviors in each of the CO, 02, C02 concentrations were observed. A model consisting of four nonlinear equations was formulated and corresponding to various combinations of parameters, oscillations in a limit cycle form were obtained. The model was used to detect oscillations obtained for this experiment as well as some observations reported earlier in the literature. 

In physics such oscillatory objects are denoted as self-sustained oscillators. Mathematically, such an oscillator is described by an autonomous (i.e., without an explicit time dependence) nonlinear dynamical system. It differs both from linear oscillators (which, if a damping is present, can oscillate only due to external forcing) and from nonlinear energy conserving systems, whose dynamics essentially depends on initial state. Dynamics of oscillators is typically described in the phase (state) space. Periodic oscillations, like those of the clock, correspond to a closed attractive curve in the phase space, called the limit cycle. The limit cycle is a simple attractor, in contrast to a strange (chaotic) attractor. The latter is a geometrical image of chaotic self-sustained oscillations. 

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