Periodically forced oscillations

Aronson, D., McGehee, R., Kevrekidis, I. Aris, R. 1986 Entrainment regions for periodically forced oscillators. Phys. Rev. A 33(3), 2190-2192. 

Fig. 16 are only rarely strictly periodic, because usually rather small fluctuations in the external parameters are sufficient to trigger abrupt changes. However, in principle, mixed-mode oscillations belong to the category of multiple-periodic limit cycles. If the behavior is governed by two incommensurate frequencies, i.e., the ratio of two periodicities is an irrational number. This situation is denoted by quasiperiodicity and has been realized experimentally with periodically forced oscillations, as will be described next. 

FIGURE 7-5 Schematic diagram of the periodic forced oscillations of a simple Hookean spring. 

Gray, S.K. (1987) A periodically forced oscillator model of van der Waals fiagmentation Classical and quantum dynamics, J.Cfien/.Pfivs- 87, 2051-2061. 

FIGURE 7.11. Time series for periodically forced oscillations in the CO oxidation on a Pt(l 10) surface exhibiting sustained oscillations [34]. (a) 1 2 Subharmonic entrainment, (b) 2 1 Superharmonic entrainment, (c) 7 2 Superharmonic entrainment, (d) Quasi-periodic response. 

A 10,0-g mass connected by a spi itig to a statiotiaiy poitit executes exactly 4 complete cycles of harmonic oscillation in 1,00 s. What are the period of oscillation, the frequency, and the angular frequency What is the force constant of the spring ... 

There is less information available in the scientific literature on the influence of forced oscillations in the control variables in polymerization reactions. A decade ago two independent theoretical studies appeared which considered the effect of periodic operation on a free radically initiated chain reaction in a well mixed isothermal reactor. Ray (11) examined a reaction mechanism with and without chain transfer to monomer. 

By cutting out a section of heated length of a Freon loop at the inlet and restoring the original flow rate, Crowley et al. (1967) found that the reduction of the heated length increased the flow stability in forced circulation with a constant power density. A similar effect was found in a natural-circulation loop (Mathisen, 1967). Crowley et al. (1967) further noticed that the change of heated length did not affect the period of oscillation, since the flow rate was kept constant. 

Recently the wall-PRISM theory has been used to investigate the forces between hydrophobic surfaces immersed in polyelectrolyte solutions [98], Polyelectrolyte solutions display strong peaks at low wavevectors in the static structure factor, which is a manifestation of liquid-like order on long lengths-cales. Consequently, the force between surfaces confining polyelectrolyte solutions is an oscillatory function of their separation. The wall-PRISM theory predicts oscillatory forces in salt-free solutions with a period of oscillation that scales with concentration as p 1/3 and p 1/2 in dilute and semidilute solutions, respectively. This behavior is explained in terms of liquid-like ordering in the bulk solution which results in liquid-like layering when the solution is confined between surfaces. In the presence of added salt the theory predicts the possibility of a predominantly attractive force under some conditions. These predictions are in accord with available experiments [99,100]. 

K. Tomita. Periodically forced nonlinear oscillators. A.V. Holden, Princenton Univ. Press. 

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. 

Let us imagine a scenario for which a supercritical Hopf bifurcation occurs as one of the parameters, fi say, is increased. For fi < fi, the stationary state is locally stable. At fi there is a Hopf bifurcation the stationary state loses stability and a stable limit cycle emerges. The limit cycle grows as ft increases above fi. It is quite possible for there to be further bifurcations in the system if we continue to vary fi. With three variables we might expect to have period-doubling sequences or transitions to quasi-periodicity such as those seen with the forced oscillator of the previous section. Such bifurcations, however, will not be signified by any change in the local stability of the stationary state. These are bifurcations from the oscillatory solution, and so we must test the local stability of the limit cycle. We now consider how to do this. 

A comparative study was done by Kevrekidis and published as I. G. Kevrekidis, L. D. Schmidt, and R. Aris. Some common features of periodically forced reacting systems. Chem. Eng. Sci. 41,1263-1276 (1986). See also two papers by the same authors Resonance in periodically forced processes Chem. Eng. Sci. 41, 905-911 (1986) The stirred tank forced. Chem. Eng. Sci. 41,1549-1560 (1986). A full study of the Schmidt-Takoudis vacant site mechanism is to be found in M. A. McKamin, L. D. Schmidt, and R. Aris. Autonomous bifurcations of a simple bimolecular surface-reaction model. Proc. R. Soc. Lond. A 415,363-387 (1988) Forced oscillations of a self-oscillating bimolecular surface reaction model. Proc. R. Soc. Lond. A 415,363-388 (1988). 

The interest in periodically forced systems extends beyond performance considerations for a single reactor. Stability of structures and control characteristics of chemical plants are determined by their responses to oscillating loads. Epidemics and harvests are governed by the cycle of seasons. Bifurcation and stability analysis of periodically forced systems is especially important in the... 

Sincic and Bailey (1977) relaxed the assumption of only one stable attractor for a given set of operating conditions and showed examples of some possible exotic responses in a CSTR with periodically forced coolant temperature. They also probed the way in which multiple steady states or sustained oscillations in the dynamics of the unforced system affect its response to periodic forcing. Several theoretical and experimental papers have since extended these ideas (Hamer and Cormack, 1978 Cutlip, 1979 Stephanopoulos et al., 1979 Hegedus et al., 1980 Abdul-Kareem et al., 1980 Bennett, 1982 Goodman et al., 1981, 1982 Cutlip et al., 1983 Taylor and Geiseler, 1986 Mankin and Hudson, 1984 Kevrekidis et al., 1984). 

In the present paper we study common features of the responses of chemical reactor models to periodic forcing, and we consider accurate methods that can be used in this task. In particular, we describe an algorithm for the numerical computation and stability analysis of invariant tori. We shall consider phenomena that appear in a broad class of forced systems and illustrate them through several chemical reactor models, with emphasis on the forcing of spontaneously oscillating systems. 

Beyond its applicability in the simplification of the computations, the stroboscopic representation greatly simplifies the recognition of patterns in the transient response of periodically forced systems. A sustained oscillation appears as a finite number of repeated points, while a quasi-periodic response appears as an invariant circle (see Figs. 3, 4, 6 and 9). 

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)],... 

A periodically forced system may be considered as an open-loop control system. The intermediate and high amplitude forced responses can be used in model discrimination procedures (Bennett, 1981 Cutlip etal., 1983). Alternate choices of the forcing variable and observations of the relations and lags between various oscillating components of the response will yield information regarding intermediate steps in a reaction mechanism. Even some unstable phase plane components of the unforced system will become apparent through their role in observable effects (such as the codimension two bifurcations described above where they collide and annihilate stable, observable responses). 

Glass, L., Guevara, M., Belair, J. and Shrier, A., 1984, Global bifurcations of a periodically forced biological oscillator. Phys. Rev. A 29,1348-1357. 

Greenspan, B. and Holmes, P., 1984, Repeated resonance and homoclinic bifurcations in a periodically forced family of oscillators. SIAM J. Math. Anal. 15, 69-97. 

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. 

Forced oscillation is a well-known technique for the characterization of linear systems and is referred to as a frequency response method in the process control field. By contrast, the response of nonlinear systems to forcing is much more diverse and not yet fully understood. In nonlinear systems, the forced response can be periodic with a period that is some integer multiple of the forcing period (a subharmonic response), or quasi-periodic (characterized by more than one frequency) or even chaotic, when the time series of the response appears to be random. In addition, abrupt transitions or bifurcations can occur between any of these responses as one or more of the parameters is varied and there can be more than one possible response for a given set of parameters depending on the initial conditions or recent history of the system. 

The centre of the forced oscillation (mean value of a2) has been chosen as a2 = 0.028 which lies in the middle of the second oscillatory region where the autonomous system has a natural period of T0 = 911.98 or frequency of... 

A feature that, to our knowledge when we discovered it, had not been seen before in forced oscillators (Marek and his co-workers have also observed it (M. Marek, personal communication)) is the folding that occurs in the left side of the 3/2 and 2/1 resonance horns. Within these folds there are two sets of stable nodes and two sets of saddles, so that bistability between the two sets exists. There are also cases of bistability between subharmonic responses of period 3 and a torus in the top of the period 3 resonance horns. In addition to the implication of bistability, the fold in the side of the 3/2 resonance horn may be of mathematical significance. Aronson et al. (1986) put forth the mathematical conjecture that if the period 3 resonance horn is a simple disc-... 

FIGURE 10 Example of chaos for AlAo 1.45, cu/stable fixed points have been found, (b) The time series for a chaotic trajectory after 150 periods of forced oscillations. The arrows indicate a near periodic solution with period 21. The periodicity eventually slips into short random behaviour followed by long near period behaviour. This near periodicity reflects the fact that the chaotic attractor forces the trajectory to eventually pass near the stable manifolds of the period 21 saddle located around the perimeter of the chaotic attractor. 

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