Oscillator limit cycle

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. 

Figure 8. Regions of different behavior of the externally driven limit cycle of Model 2 (see Figures 5-7) (I) free oscillation (limit cycle with (II) quasi-peri-odic oscillation (III) entrainment (limit cycle with X) (IV) unstable region — onset of traveling waves.

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

A dynamic system may exhibit qualitatively different behavior for different values of its control parameters 0. Thus, a system that has a point attractor for some value of a parameter may oscillate (limit cycle) for some other value. The critical value where the behavior changes is called a bifurcation point, and the event a bifurcation [32]. More specifically, this kind of bifurcation, i.e., the transition from a point attractor to a limit cycle, is referred to as Hopf bifurcation. 

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... 

Mathematical models of the reaction yield various solutions. Some of the solutions obtained are One singular point, 3 singular points, oscillating limit cycle, double periodic oscillations, chaotic oscillations. 

Such sustained oscillations (limit cycles) of the temperature and the conversion are mostly due to a unique steady state solution of Eq. (1) which is unstable to small perturbations. The region of parameter space for which instabilities occur can be plotted into a so-called stability diagram. Fig. 1 gives y versus u with Dao as a fixed parameter. The curves a = o and B = 0 calculated from Eqs. (7) and (8) are drawn into this diagram. It will be used here to present the results for a lot of numerical calculations concerning limit cycles in the region a > 0, B > 0. 

Recently a new reaction scheme the "Explodator core was proposed by us [1,2] as a simple model for chemical oscillations. Limit cycle oscillators can be constructed using that e3q>losive core and different limitary reactions. 

One unique but normally undesirable feature of continuous emulsion polymerization carried out in a stirred tank reactor is reactor dynamics. For example, sustained oscillations (limit cycles) in the number of latex particles per unit volume of water, monomer conversion, and concentration of free surfactant have been observed in continuous emulsion polymerization systems operated at isothermal conditions [52-55], as illustrated in Figure 7.4a. Particle nucleation phenomena and gel effect are primarily responsible for the observed reactor instabilities. Several mathematical models that quantitatively predict the reaction kinetics (including the reactor dynamics) involved in continuous emulsion polymerization can be found in references 56-58. Tauer and Muller [59] developed a kinetic model for the emulsion polymerization of vinyl chloride in a continuous stirred tank reactor. The results show that the sustained oscillations depend on the rates of particle growth and coalescence. Furthermore, multiple steady states have been experienced in continuous emulsion polymerization carried out in a stirred tank reactor, and this phenomenon is attributed to the gel effect [60,61]. All these factors inevitably result in severe problems of process control and product quality. 

A little later (1929) the Russian physicist Andronov pointed out3 that the stationary state of self-excited oscillations discovered by van der Pol is expressible analytically in terms of the limit cycle concept of the theory of PoincarA... 

However, this simple model of a periodic motion occupied the central position in the theory of oscillations from its very beginning (Galileo) up to the time of Poincar6, when it was replaced by the new model—the limit cycle. 

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. 

As a closed trajectory in the phase plane means obviously a periodic phenomenon, the discovery of limit cycles was fundamental for the new theory of self-excited oscillations. 

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. 

The reader can recognize that a stable limit cycle illustrates the definition (5) while a harmonic oscillator illustrates that given by (4). 

Achieving steady-state operation in a continuous tank reactor system can be difficult. Particle nucleation phenomena and the decrease in termination rate caused by high viscosity within the particles (gel effect) can contribute to significant reactor instabilities. Variation in the level of inhibitors in the feed streams can also cause reactor control problems. Conversion oscillations have been observed with many different monomers. These oscillations often result from a limit cycle behavior of the particle nucleation mechanism. Such oscillations are difficult to tolerate in commercial systems. They can cause uneven heat loads and significant transients in free emulsifier concentration thus potentially causing flocculation and the formation of wall polymer. This problem may be one of the most difficult to handle in the development of commercial continuous processes. 

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... 

If the roots are pure imaginary numbers, the form of the response is purely oscillatory, and the magnitude will neither increase nor decay. The response, thus, remains in the neighbourhood of the steady-state solution and forms stable oscillations or limit cycles. 

McCarley, R. W. Massaquoi, S. G. (1986). A limit cycle mathematical model of the REM sleep oscillator system. Am. J. Physiol 251, R1011-R29. 

Self-sustained Oscillations. Under certain conditions, isothermal limit cycles in gaseous concentrations over catalysts are observed. These are probably caused by interaction of steps on the surface. Sometimes heat and mass transfer effects intervene, leading to temperature oscillations also. Since this subject has recently been reviewed (42, 43) only a few recent papers will be mentioned here. 

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. 

The experiments and the simulation of CSTR models have revealed a complex dynamic behavior that can be predicted by the classical Andronov-Poincare-Hopf theory, including limit cycles, multiple limit cycles, quasi-periodic oscillations, transitions to chaotic dynamic and chaotic behavior. Examples of self-oscillation for reacting systems can be found in [4], [17], [18], [22], [23], [29], [30], [32], [33], [36]. The paper of Mankin and Hudson [17] where a CSTR with a simple reaction A B takes place, shows that it is possible to drive the reactor to chaos by perturbing the cooling temperature. In the paper by Perez, Font and Montava [22], it has been shown that a CSTR can be driven to chaos by perturbing the coolant flow rate. It has been also deduced, by means of numerical simulation, that periodic, quasi-periodic and chaotic behaviors can appear. 

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