Subharmonic oscillator

If at is an integer, say a> = n we have rather complicated manifestations of the subharmonic resonance. If it is at n + e, where e is a certain small number, one has still the subharmonic resonance, but it is accompanied by another phenomenon of synchronization, which con-, gists in the entrainment of the frequency of the avtoperiodic oscillation (if A = 0), by that of the heteroperiodic oscillation (the externally applied one). 

Assume that we have a pendulum (Fig. 6-14) provided with a piece of soft iron P placed coaxially with a coil C carrying an alternating current that is, the axis of the coil coincides with the longitudinal axis OP of the pendulum at rest. If the coil is excited, one finds that the pendulum in due course begins to oscillate, and th oscillations finally reach a stationary amplitude. It is important to note that between the period of oscillation of the pendulum and the period of the alternating current there exists no rational ratio, so that the question of the subharmonic effect is ruled out. 

In a series of experiments we have tested the type and range of entrainment of glycolytic oscillations by a periodic source of substrate realizing domains of entrainment by the fundamental frequency, one-half harmonic and one-third harmonic of a sinusoidal source of substrate. Furthermore, random variation of the substrate input was found to yield sustained oscillations of stable period. The demonstration of the subharmonic entrainment adds to the proof of the nonlinear nature of the glycolytic oscillator, since this behavior is not observed in linear systems. A comparison between the experimental results and computer simulations furthermore showed that the oscillatory dynamics of the glycolytic system can be described by the phosphofructokinase model. 

It is interesting to consider the shapes of the subharmonic trajectories that lock on the torus in the various entrainment regions of order p/q. The subharmonic period 4 at the 4/3 resonance horn is, for example, a three-peaked oscillation in time [Fig. 7(a)] and has three closed loops in its phase-plane projection [Fig. 7(b)], while the subharmonic period 4 at the 4/ 1 resonance is a single-peaked, single-loop oscillation [Figs. 7(d) and 7(e)]. A subharmonic period 2 at the 2/3 resonance is also included in Figs. 7(g) and 7(h). Multipeaked oscillations observed in chemical systems (Scheintuch and Schmitz, 1977 Flytzani-Stephanopoulos et al., 1980) may thus result from the interaction of frequencies of local oscillators. Such trajectories are the nonlin-... 

FIGURE 7 Typical shapes of subharmonic trajectories. A subharmonic period 4 within the 4/3 resonance horn is a three-peaked oscillation in time (a), has three loops in its phase plane projection (b), and four loops in its x-cos 0 projection (c) (Brusselator, a = 0.0072, o = 4/3). The subharmonic period 4 within the 4/1 resonance horn has one loop in its phase plane projection (e), four loops in the x-cos projection (f) and is a one-peaked oscillation in time (d). Stroboscopic points are denoted by O. Try to imagine them winding around the doughnut in three-dimensional space An interesting shape shows up at the period 2 resonance in the 2/3 resonance horn (surface model >/aio = 2/3, alao = 0.1, o0 = 0.001) (g, h). These shapes are comparatively simple because of the shape of the unperturbed limit cycle which for all cases was a simple closed curve. 

At interesting phenomenon occurs in the case of other resonance horns we have studied it for the case of the 3/1 resonance. The torus pattern breaks when the subharmonic periodic trajectories locked on it for small FA decollate from the torus as FA increases. We are left then with two attractors a stable period 3 and a stable quasi-periodic trajectory. This is a spectacular case of multistability (co-existence of periodic and quasi-periodic oscillations). The initial conditions will determine the attractor to which the system will eventually converge. This decollation of the subharmonics from the torus was predicted by Greenspan and Holmes (1984). They also predicted chaotic trajectories close to the parameter values where the subharmonic decollation occurs. 

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. 

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-... 

Capsaskis, S. C. Kenney, C. N. 1986 Subharmonic response of a heterogeneous catalytic oscillator, the Cantabrator , to a periodic input. J. Phys. Chem. 90,4631-4637. 

Fig. 12.2 (a) Experimental recording of the proximal tubular pressure in a single nephron of a normotensive rat. The power spectrum (b) clearly shows the TGF-mediated oscillations at fsiow(t) 0.034 Hz and the myogenic oscillations at ffasi(t) 0.16 Hz. The spectrum also displays harmonics and subharmonics of the TGF-oscillations. A plot... 

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.

Fig. 17. Experimental dynamic phase diagram for forced oscillations in the CO/O, reaction on Pt(l 10). Periodic modulation of the 02 pressure with amplitude A (as a percentage of the constant base pressure) and period length, Tex, with respect to that of the autonomous oscillations, T . (From Ref. 91.) Fixed parameters, for the subharmonic (superharmonic) range p(h = 3.0 (4.15) x 10 5 torr, p , = 1.6 (2.1) x 10 5 torr, T = 525 (530) K.

Direct evidence for a spin density wave transport is the detection of a current oscillating at a frequency that is proportional to the dc current carried collectively. The recent observation of such oscillations the harmonic and subharmonic locking of this oscillation to an external ac source and a motional narrowing of the NMR spectrum in the sliding SDW state have established firm evidence for the existence of a novel collective transport in a SDW condensate. 

Figure 35 (a) Detection of a beating between the ac current provided by an external source and the internal current oscillation in the sliding SDW state (b) linear dependence of the SDW current against the frequency of the internal current oscillation, harmonics, and subharmonics. (After Ref. 110.)... 

The response to external forcing with frequency and amplitude A may be classified as follows [31-33] If the resulting period Tj. of the system exhibit a fixed phase relation to that of the modulation Tex, the system is entrained. The ratio Tr/Tex may be expressed as that between two small numbers, that is, Tr/Tex = k/l. For k/l =, the entrainment is called harmonic, for k/l> super harmonic, and for k/lphase difference between response and modulation varies continuously, the oscillations are called quasi-periodic. 

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. 

Dipole oscillations in an assembly of molecules in the membrane of cells can be modeled as phase-locked solid state oscillators by a basic circuit as in Figure 1. Loose coupling between such circuits imposes an eigenvalue problem from which significant mode softening can be shown to result and this has been suggested to be an important requirement in the energetics associated with the reproduction and mutation of cells. As each individual unit oscillator can operate at subharmonics as well as harmonics, the above model is consistent with the idea that in vivo a number of discrete frequencies exist in the cell. 

In an empty collapsing bubble, the amplitude of the spherical harmonic components increases proportionally to and oscillates with increasing frequency as R tends towards zero. This is a situation for which the sinklike nature of the flow gets the better of stabilization due to acceleration. In the case of a gas bubble, the situation is quite different, however. The problem, approched independently by Benjamin, 2 h s received considerable attention from experimentalists in order to quantify the heat transfer, to link the non-radial instability and subharmonics (acoustic) emission,determine the threshold for triggering off these instabilities. ... 

Thompson J.M.T., Bokaian A.R. and Ghaffari R., Subharmonic resonances and chaotic motions of a bilinear oscillator. J. Appl. Math., 31, 207-234 (1983)... 

Figure 12 depicts the time correlation function of the variable x. After a stage of rapid decrease we observe long-living irregular oscillations around zero suggesting that the attractor is organized around a noisy periodic or multi-periodic solution. This is further corroborated by the power spectrum, which features a number of pronounced peaks around a principal frequency w 1.2 and its first few harmonics and subharmonics. 

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