Quasiperiodic oscillations

The solutions of these equations (the trajectories) will for long times (i.e., after transient effects associated with switching on the external parameters have decayed) approach so-called limit sets, which may be classified into fixed points (stationary states), limit cycles (periodic oscillations), mixedmode oscillations, quasiperiodic oscillations, and chaotic behavior. Transitions between these states may occur upon variation of the external parameters pk and are called bifurcations. Experimental evidence for these effects with the system CO + 02/Pt(110) will be briefly presented without going further into details of the underlying general theory (see 16, 17). 

Fig. 1. Some examples of heterogeneous oscillations observed experimentally, (a) Sinusoidal time series for H2/O2 on a Pt wire (from Ref. 41). (b) Relaxation oscillations for CO/O2 on Pt/Al203 (from Ref. 100). (c) Oscillations after a single period doubling for CO/O2 on Pt(l 10) (from Ref. 231). (d) Model and experimental (inset) quasiperiodic oscillations for NO/CO on supported Pd (from Ref. 232). (e) Deterministic chaos produced by a period doubling sequence for CO/O2 on Pt(IlO) (from Ref. 231).

Now we apply all this to elucidate the structure of accretion disks near Schwarzschild BHs. Ignoring fluid and magnetohydrodynamic effects, we consider free particles in circular orbits on the equatorial plane of the Schwarzschild metric. A key question is what is the relation between the azimuthal frequency ua and the radius r of the orbit. For example, the frequencies of the quasiperiodic oscillations (QPOs) seen in some galactic X-ray sources are thought to reflect the va of close-in orbits. 

Periodic and Quasiperiodic Oscillations for Systems with Lag. Vyshcha Shkola,Kiev, 1979. 

Figure 8.3. Quasiperiodic oscillation regime for the reaction intermediates according to the Oregonator kinetic model.

Figure 8.5 demonstrates the calculated phase diagram for the Oregonator kinetie model, describing the dynamic behavior of the system in coordinates of the initial concentrations [A]o and [Y]o. The phase diagram comprises the zones of dilferent reaction modes for the quasiperiodic oscillations, the stationary mode and the ehaotic damped oseillations. 

Fig. 13.6, the radiating and guiding components are split off near the focal point shown as a small circle. The interference between these components gives rise to the oscillations of the evanescent field and to the appearance of quasiperiodic dips, which are clearly seen in Fig. 13.7. 

SUMMARY We investigate the unsteady motion of mass reservoir formed by the accretion onto the magnetosphere around rotating neutron stars. The unsteady motion of the reservoir induces secondary accretion to neutron star by R-T instability. The nonperiodic or quasiperiodic phenomena of X-ray bursters seems to be related to this property of mass reservoir on the magnetosphere. We classify the typical dynamical state of the reservoir into three types with the parameters which are accretion rate M and angular velocity of neutron star ft. They are nonsequential oscillation sequential periodic (quasi-periodic) oscillation, and chaotic oscillation states. 

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. 

Fig. 21. Theoretical dynamic phase diagram for forced oscillations on Pt(l 10), evaluated from Eqs. (4), (5a), and (6). The calculations have been performed up to larger forcing amplitudes than were experimentally accessible (compare with Fig. 17). For small A, the bands characterizing entrainment and quasiperiodicity are discernible, whereas for larger A more complex bifurcations result. Types of bifurcations ns, Neimark-Sacker pd, period doubling snp, saddle-node of periodic orbits. (From Ref. 69.)...

Badii, R. and Meier, P.F. (1987). Comment on Chaotic Rabi oscillations under quasiperiodic perturbations , Phys. Rev. Lett. 58, 1045. 

Pomeau, Y., Dorizzi, B. and Grammaticos, B. (1986). Chaotic Rabi oscillations imder quasiperiodic perturbation, Phys. Rev. Lett. 56, 581-684. 

In a review article on oscillatory reactions (294), Sheintuch discusses the effect of introducing a heat balance for the catalyst rather than a mass balance for the reactor into the differential equation system for a surface reaction with oxidation/reduction cycles. Although the coverage equations alone can yield oscillatory behavior, as was the case for the models discussed in the previous section, Sheintuch s model is discussed in this section because introduction of the heat balance adds qualitatively new features. In this extended system complex, multiple peak behavior and quasiperiodicity was observed as shown in Fig. 8. Sheintuch also investigated the interaction of two oscillators. This work, however, will be treated in detail in Section V, were synchronization and chaos are discussed. 

The power sjjectrum is merely the Fourier transform of the VAF via the Wiener-Khintchine theorem. The integration is carried out as a discrete sum over the jjeriod of time in which the VAF decays to a zero value. This quantity gives the number of oscillators at a given frequency and is a very informative indicator of the transition from rigid, quasiperiodic motion to nonrigid, chaotic motion. Note that I(co = 0) is proportional to the diffusion constant. This quantity was calculated by Dickey and Paskin - in the study of phonon frequencies in solids and also by Kristensen et al. in simulations of cluster melting. 

Explaining Lissajous figures) Lissajous figures are one way to visualize the knots and quasiperiodicity discussed in the text. To see this, consider a pair of uncoupled harmonic oscillators described by the four-dimensional system. r-)-x = 0, y + co y Q. 

The central region of the local mode representation of the phase space trajectories is called the resonance region. In this resonance region of phase space the trajectories ( 3 and 4) are not free to explore the full 0 < ip < n range and are threfore classified as normal mode trajectories. Points A and B are fixed points which he at the maximum and minimum E extremes, Ea(I) and Eb(I), of the resonance region for a particular value of I. Point A at Iz = 0 (vr = vi) and ip = 7t/2 (out-of-phase motion of the R and L oscillators) is stable and corresponds to a pure antisymmetric stretch. Point B at Iz = 0 (vr = vl) and ip = 0 and 7r (in-phase motion) is unstable (because it lies on the separatrix) and corresponds to a pure symmetric stretch. Quasiperiodic trajectories that circulate about a stable fixed point resemble the fixed point periodic trajectory. At E > Ea(I) no trajectories of any type can exist. At E < Er(I) the B-like trajectories vanish and are replaced by trajectories that circulate about the Ca, Cb fixed points and are therefore C-like. The Ca and Cb lines (Iz = / = 2, 0 < ip < 7r) are actually the north and south poles on the local mode polyad phase sphere (Fig. 9.13(c)). The stable fixed points he near the poles and trajectories la and 2a circulate about the fixed point near Ca and trajectories lb... 

More testing of these approaches is certainly required before any final pronouncements on their worth can be made. One important contribution to assessing the ideas was that of Sewell et al. (53). For a system of coupled oscillators, they showed that the constraints of Miller et al. and Bowman et al. (45,46) may induce physically undesirable effects. In particular, they gave an example of a trajectory in the quasiperiodic regime being transformed into a chaotic one by the action of the constraints. 

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