Oscillations quasi periodic

Since the first report of oscillation in 1965 (159), a variety of other nonlinear kinetic phenomena have been observed in this reaction, such as bi-stability, bi-rhythmicity, complex oscillations, quasi-periodicity, stochastic resonance, period-adding and period-doubling to chaos. Recently, the details and sub-systems of the PO reaction were surveyed and a critical assessment of earlier experiments was given by Scheeline and co-workers (160). This reaction is beyond the scope of this chapter and therefore, the mechanistic details will not be discussed here. Nevertheless, it is worthwhile to mention that many studies were designed to explore non-linear autoxidation phenomena in less complicated systems with an ultimate goal of understanding the PO reaction better. 

In Be stars, the uniform rotation have been shown to be unstable in the presence of a prograde and a retrograde mode. Ultimately their rotation profile has been pointed out by Ando(1986) to oscillate quasi-periodically around the uniform rotation (see Ando(1986) for the detailed explanation of the oscillatory behavior of the rotation profile). 

As // increases further, the system may show more and more complicated dynamics through a number of bifurcations. It may show complicated periodic oscillations, quasi-periodic oscillations or a variety of non-periodic behaviors. For instance, we know of the recent discoveries of fantastic bifurcation structures in the spatially homogeneous Belousov-Zhabotinsky reaction, see Hudson et al., 1979. 

The next problem to consider is how chaotic attractors evolve from tire steady state or oscillatory behaviour of chemical systems. There are, effectively, an infinite number of routes to chaos [25]. However, only some of tliese have been examined carefully. In tire simplest models tliey depend on a single control or bifurcation parameter. In more complicated models or in experimental systems, variations along a suitable curve in the control parameter space allow at least a partial observation of tliese well known routes. For chemical systems we describe period doubling, mixed-mode oscillations, intennittency, and tire quasi-periodic route to chaos. 

In addition to tire period-doubling route to chaos tliere are otlier routes tliat are chemically important mixed-mode oscillations (MMOs), intennittency and quasi-periodicity. Their signature is easily recognized in chemical experiments, so tliat tliey were seen early in the history of chemical chaos. 

Figure 3. Constraints from orbital frequencies. The 1330 Hz curve is for the highest kilohertz quasi-periodic oscillation frequency yet measured (for 4U 0614+091, by van Straaten et al. 2000). The 1500 Hz curve shows a hypothetical constraint for a higher-frequency source. Other lines are as in Figure 1. All curves are drawn for nonrotating stars the constraint wedges would be enlarged slightly for a rotating star (see Miller, Lamb, Psaltis 1998).

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. 

L.E. Myers, R.C. Eckardt, M.M. Fejer, R.L. Byer, and W.R. Bosenberg, "Multigrating quasi-phase-matched optical parametric oscillator in periodically poled LiNb03, Optics Letters 21, 591 (1996). 

If the quotient o>/a>0 is irrational, the path across the toroidal surface will return to a different point on the completion of each cycle. Eventually the trajectory will pass over every point on the surface of the torus without ever forming a closed loop. This is quasi-periodicity , and an example is shown in Fig. 13.11. The corresponding concentration histories do not necessarily give complex waveforms, as can be seen from the figure. However, the period of the oscillations is neither simply that of the natural cycle nor just that of the forcing term, but involves both. 

For sufficiently large forcing amplitudes the oscillation becomes completely entrained, with a period exactly equal to one forcing period, whatever that value of a>/a>0. The entrainment may arise from a phase-locked response—as seen previously in Fig. 13.9—or from a quasi-periodic pattern. The boundary for full entrainment appears as an almost straight line with positive slope of oj/oj0 > 1 and negative slope for oj/oj0 < 1. 

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. 

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. 

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). 

A quasi-periodic solution to a system of ODEs is characterized by at least two frequencies that are incommensurate (their ratio is an irrational number) (Bohr, 1947 Besicovitch, 1954). Several such frequencies may be present on high-order tori, but for the two-dimensional forced systems we examine, we may have no more than two distinct frequencies (a two-torus, T2). A quasi-periodic solution is typically bora when a pair of complex conjugate FMs of a periodic trajectory leave the unit circle at some angle , where /2ir is irrational. Such a solution is also expected when we periodically perturb an autonomously oscillating system with a frequency incommensurate to its natural frequency. 

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. 

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. 

Quasi-periodic variability of climate (its most vivid manifestations are quasibiennial oscillations in the equatorial stratosphere). 

El Nino/Southern Oscillation (in view of a wide range of frequencies this event cannot be considered quasi-periodic). 

In an open system such as a CSTR chemical reactions can undergo self-sustained oscillations even though all external conditions such as feed rate and concentrations are held constant. The Belousov-Zhabotinskii reaction can undergo such oscillations under isothermal conditions. As has been demonstrated both by experiments [1] and by calculations 12,3] this reaction can produce a variety of oscillation types from simple relaxation oscillations to complicated multipeaked periodic oscillations. Evidence has also been given that chaotic behavior, as opposed to periodic or quasi-periodic behavior, can take place with this reaction [4-12]. In addition, it has been shown in recent theoretical studies that chaos can occur in open chemical reactors [11,13-17]. 

It is simplest to think of sharkskin as a result of a quasi-periodic perturbation on the overall extrudate swell. This small amplitude fluctuation of extrudate swell arises from the oscillation of the boundary condition at the exit wall that produces an oscillation of the local stress level as the interfacial chains suffer a conformational instability. The local boundary condition oscillates between noslip and slip, resulting in the fluctuation of the stress level at the die exit. To determine whether some sort of melt fracture occurs, we need to know not only the... 

The term melt fracture has been applied from the outset [9,13] to refer to various types of visible extrudate distortion. The origin of sharkskin (often called surface melt fracture ) has been shown in Sect. 10 to be related to a local interfacial instability in the die exit region. The alternating quasi-periodic, sometimes bamboo-like, extrudate distortion associated with the flow oscillation is a result of oscillation in extrudate swell under controlled piston speed due to unstable boundary condition, as discussed in Sect. 8. A third type, spiral like, distortion is associated with an entry flow instability. The latter two kinds have often been referred to as gross melt fracture. It is clearly misleading and inaccurate to call these three major types of extrudate distortion melt fracture since they do not arise from a true melt fracture or bulk failure. Unfortunately, for historical reasons, this terminology will stay with us and be used interchangeably with the phase extrudate distortion. ... 

We also now know that complex oscillations evolve as simple limit cycles become unstable, bifurcating to more complex limit cycles. Only a small number of bifurcation sequences account for all known scenarios. We have seen examples of mixed-mode sequences (H2 -I- O2) and period-doubling cascades (CO -I- O2). A third route involving quasi-periodic responses is known and arises in some chemical system [88], but has not yet been observed in combustion systems (except in some special studies in which the ambient temperature or some other parameter is forced to vary in some sinusoidal or other periodic manner [89]). The important lesson then... 

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