Limit cycle frequency

The model predicts rate oscillations over a rather wide range of parameters A. It was verified that for given feed composition, space velocity and reactor temperature the predicted limit cycle frequency and amplitude do not depend on the initial conditions of the numerical integration. 

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). 

If this expected photoemission really takes place, the resultant spectra should reflect the nonhnear dynamics of nonadiabatic vibrational motion under an external field, which is similar to classical driven oscillators such as a forced Duffing oscillator [156, 239]. Therefore various nonlinear phenomena such as limit cycle, frequency locking, and chaos (1.5-dimensional chaos) [156, 239] can be expected, which would be intrinsically originated from the quantmn dynamics. Furthermore, one may be able to control the frequency and amplitude of the photoemission by varying the laser parameters applied. It may be possible to utilize the photoemission as a new optical somce and also as finger-print signals to identify molecular species and/or molecular states. In this section we illustrate the appearance of such... 

The quasiperiodic route to chaos is historically important. It arises from a succession of Hopf birfurcations. As already noted, a single Hopf bifurcation results in a limit cycle. The next Hopf bifurcation produces a phase flow tliat can be represented on tire surface of a toms (douglmut). This flow is associated witli two frequencies if tire ratio of tliese frequencies is irrational tlien tire toms surface is densely covered by tire phase trajectory, whereas if... 

The opposite effect is depicted on Fig. 8.32 where the catalyst under open-circuit conditions exhibits stable limit cycle behaviour with a period of 184 s. Imposition of a negative current of -400 pA leads to a steady state. Upon current interruption the catalyst returns to its initial oscillatory state. Application of positive currents leads to higher frequency oscillatory states. 

The period of the limit cycle is the ultimate period (PJ for the transfer function relating the controlled variable x and the manipulated variable m. So the ultimate frequency is... 

This method can be extended to evaluate models with more parameters and with diHerent kinds of transfer functions (e.g., underdamped second-order lag) by using hysteresis in the relay feedback or by inserting an additional dead time in the loop to produce a limit cycle with a different frequency. The two autotune tests give four equations so four parameters can be evaluated. 

The frequency of modulation il is now the main parameter, and we are able to switch the system of SHG between different dynamics by changing the value of il. To find the regions of where a chaotic motion occurs, we calculate a Lyapunov spectrum versus the knob parameter il. The first Lyapunov exponent A,j from the spectrum is of the greatest importance its sign determines the chaos occurrence. The maximal Lyapunov exponent Xj as a function of is presented for GCL in Fig. 6a and for BCL in Fig. 6b. We see that for some frequencies il the system behaves chaotically (A-i > 0) but orderly Ck < 0) for others. The system in the second case is much more damped than in the first case and consequently much more stable. By way of example, for = 0.9 the system of SHG becomes chaotic as illustrated in Fig. 7a, showing the evolution of second-harmonic and fundamental mode intensities. The phase point of the fundamental mode draws a chaotic attractor as seen in the phase portrait (Fig. 7b). However, the phase point loses its chaotic features and settles into a symmetric limit cycle if we change the frequency to = 1.1 as shown in Fig. 8b, while Fig. 8a shows a seven-period oscillation in intensities. To avoid transient effects, the evolution is plotted for 450 < < 500. 

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). 

We now have a total of six parameters four from the autonomous system (p, r0, and the desorption rate constants k, and k2) and two from the forcing (rf and co). The main point of interest here is the influence of the imposed forcing on the natural oscillations. Thus, we will take just one set of the autonomous parameters and then vary rf and co. Specifically, we take p = 0.019, r0 = 0.028, fq = 0.001, and k2 = 0.002. For these values the unforced model has a unique unstable stationary state surrounded by a stable limit cycle. The natural oscillation of the system has a period t0 = 911.98, corresponding to a natural frequency of co0 = 0.006 889 6. 

Fig. 13.9. The forced Takoudis-Schmidt-Aris model with a forcing frequency twice that of the natural oscillation (a) zero-amplitude forcing (autonomous oscillation and limit cycle) (b) r, = 0.002 (c) r, = 0.003 (d) r, = 0.004 (e) r, = 0.005 (f) rr = 0.006 (g) rf = 0.007 (h) rf = 0.01. Traces show the time series 0p(t) over 10 natural periods (or 20 forcing periods) and the associated limit cycle in the 0 -6, plane.

In the previous subsection, the forcing frequency was exactly twice the natural oscillatory frequency. Thus the motion around one oscillation gives exactly two circuits of the forcing cycle for one revolution of the natural limit cycle. The full oscillation of the forced system has the same period as the autonomous cycle and twice the forcing period. The concentrations 0p and 6r return to exactly the same point at the top of the cycle, and subsequent oscillatory cycles follow the same close path across the toroidal surface. This is known as phase locking or resonance. We can expect such locking, with a closed loop on the torus, whenever the ratio of the natural and forcing... 

In between the resonance horns are regions of the parameter plane for which the response is quasi-periodic. Note that it is even possible for the frequencies to have a simple ratio and yet for the system to lie outside the corresponding resonance horn if the amplitude is raised. Figure 13.15 shows two time series for forcing with oj/oj0 = 10/1. At low forcing amplitude, rr = 0.005, we have phase locking and a simple if rather crumpled limit cycle. With rf = 0.01, however, the response is quasi-periodic a few cycles are shown and demonstrate quite well how the trajectory begins to wind around the torus. 

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. ... 

When the forcing amplitude is very small and the midpoint of the forcing oscillation scans the autonomous bifurcation diagram, the qualitative response of the forced system for all frequencies can be deduced from the autonomous system characteristics. As the amplitude of the forcing becomes larger, one cannot predict a priori what will occur for a particular system. For this example, the most complicated phenomenon possible is a turning point bifurcation on a branch of periodic solutions where two limit cycles, one stable and one unstable, collide and disappear. This will appear as a pinch on the graph of the map [Fig. 1(d)],... 

The effects of forced oscillations in the partial pressure of a reactant is studied in a simple isothermal, bimolecular surface reaction model in which two vacant sites are required for reaction. The forced oscillations are conducted in a region of parameter space where an autonomous limit cycle is observed, and the response of the system is characterized with the aid of the stroboscopic map where a two-parameter bifurcation diagram for the map is constructed by using the amplitude and frequency of the forcing as bifurcation parameters. The various responses include subharmonic, quasi-peri-odic, and chaotic solutions. In addition, bistability between one or more of these responses has been observed. Bifurcation features of the stroboscopic map for this system include folds in the sides of some resonance horns, period doubling, Hopf bifurcations including hard resonances, homoclinic tangles, and several different codimension-two bifurcations. 

In the limiting case of very large-amplitude forcing, the system is completely dominated by the forcing and the only response possible is a stable limit cycle of period 1 (same period as the forcing). The amplitude at which this becomes the case varies greatly and seems to increase with the forcing frequency. 

For an exothermic decomposition of N20 Hugo (37, 47) observed limit cycles with a frequence = 3 min and amplitudes — 30°C. 

For a given damping (T = 0.025) the amplitude and the frequency of the driving force were chosen so that the chaotic attractor coexists with the stable limit cycle (SC) h = 0.13,0)/ = 0.95 (see Fig. 13). 

The available data from emulsion polymerization systems have been obtained almost exclusively through manual, off-line analysis of monomer conversion, emulsifier concentration, particle size, molecular weight, etc. For batch systems this results in a large expenditure of time in order to sample with sufficient frequency to accurately observe the system kinetics. In continuous systems a large number of samples are required to observe interesting system dynamics such as multiple steady states or limit cycles. In addition, feedback control of any process variable other than temperature or pressure is impossible without specialized on-line sensors. This note describes the initial stages of development of two such sensors, (one for the monitoring of reactor conversion and the other for the continuous measurement of surface tension), and their implementation as part of a computer data acquisition system for the emulsion polymerization of methyl methacrylate. 

Most elastic elements will have a service life of close to a million cycles, if the cycle time is not less than 5 seconds. If longer service life or higher-cycle frequencies are required, metal fatigue tends to limit the usefulness of elastic elements, and special designs, such as sealed piston elements, should be considered. 

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. 

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