Oscillation oscillatory regime

The solution of the first kind is stable and arises as the limit, t —> oo, of the non-stationary kinetic equations. Contrary, the solution of the second kind is unstable, i.e., the solution of non-stationary kinetic equations oscillates periodically in time. The joint density of similar particles remains monotonously increasing with coordinate r, unlike that for dissimilar particles. The autowave motion observed could be classified as the non-linear standing waves. Note however, that by nature these waves are not standing waves of concentrations in a real 3d space, but these are more the waves of the joint correlation functions, whose oscillation period does not coincide with that for concentrations. Speaking of the auto-oscillatory regime, we mean first of all the asymptotic solution, as t —> oo. For small t the transient regime holds depending on the initial conditions. 

These models consider the mechanisms of formation of oscillations a mechanism involving the phase transition of planes Pt(100) (hex) (lxl) and a mechanism with the formation of surface oxides Pd(l 10). The models demonstrate the oscillations of the rate of C02 formation and the concentrations of adsorbed reactants. These oscillations are accompanied by various wave processes on the lattice that models single crystalline surfaces. The effects of the size of the model lattice and the intensity of COads diffusion on the synchronization and the form of oscillations and surface waves are studied. It was shown that it is possible to obtain a wide spectrum of chemical waves (cellular and turbulent structures and spiral and ellipsoid waves) using the lattice models developed [283], Also, the influence of the internal parameters on the shapes of surface concentration waves obtained in simulations under the limited surface diffusion intensity conditions has been studied [284], The hysteresis in oscillatory behavior has been found under step-by-step variation of oxygen partial pressure. Two different oscillatory regimes could exist at one and the same parameters of the reaction. The parameters of oscillations (amplitude, period, and the... 

Fig. 37. Gray-scale representation of a calculated spatio-temporal evolution of the potential at a ring electrode (top) and time series of the global current (bottom) in the oscillatory regime of the N-NDR oscillator. (The calculation was done for reduction currents hence, the largest current is in the peak of the oscillation.)...

For the CO oxidation reaction on Pt, in particular, several fundamental questions still remain unanswered. Quite recently doubts and questions have been raised even about the existence of an oscillatory regime for this reaction system. Cutlip and Kenney [l2] have been unable to observe any oscillations during the oxidation of CO over a 0.5% Pt/y-Al203 catalyst in a recycle reactor. Their study, however, utilized low feed compositions of CO (0.5-3%)... 

Otterstedt et a/.also studied waves in the oscillatory regime during Co dissolution. The oscillations possess a relaxationhke character, which is typical for oscillations between the active and the passive state of metal dissolution reactions. They are characterized by long, quasi-stationary periods of vanishing current density, followed by a sharp... 

Oxidation of methane in the presence of such a binary oxide-metal catalyst proceeds in an oscillatory regime, and both temperature and concentration oscillations take place. Oscillations arise at the temperature at which the rate of reaction over the oxide component becomes noticeable ( 500°C). As temperature increases, the oscillation amplitude passes through a maximum. The oscillatory behavior disappears when complete conversion of oxygen is reached. In other words, the range of temperatures in which the oscillations are observed covers the range of oxygen conversions from 0 to 100%. 

In the middle row the oscillatory regime is illustrated. The systems exhibits continuous oscillations. Perturbations have at this stage little influence on the dynamics. 

The field of Marangoni instabilities shows a large variety of dissipative structures, including the principle of stationary structures, hierarchical structures with limited self-similarity, relaxation oscillations and regular behavior of travelling autowaves with chaotic turbulence-like behaviour. There is also the oscillatory regime with trains of waves with soliton-like behaviour of each wave. Anormal as well as normal dispersion of these waves have recently... 

That sustained oscillations in the mitotic cascade correspond to the evolution toward a limit cycle in the (C, M, X) space is demonstrated in fig. 10.7. Indeed, for a given set of parameter values the system always reaches the same oscillatory regime characterized by a fixed amplitude and frequency, regardless of initial conditions. 

Within this restrictive framework of two-variable models, Albert Goldbeter derives fascinating original results such as birhythmicity, which allows a system to choose between two simultaneously stable oscillatory regimes. With the number of variables, the repertoire of dynamic phenomena increases rapidly. Now, besides simple periodic behaviour we can also predict and observe complex oscillations of the bursting type, the coexistence between more than two rhythms, or the evolution toward chaos. As the author shows, small variations in the values of some control parameters permit the switch from one mode of behaviour to the other. The essential elements, in all cases, are the feedback mechanisms of biochemical reactions and the fact that these reactions occur far from equilibrium. 

Consider an oscillatory reaction run in a CSTR. It is sufficient that one species in the system is measured. A delayed feedback related to an earlier concentration value of one species is applied to the inflow of a species in the oscillatory sysy tern. The feedback takes the form Xjoit) = f Xi[t - r)), where the input concentration Xjoit) oftheyth species at time t depends on Z, (t - r ), the instantaneous concentration of the ith species at time t - r. Once the delayed feedback is applied, the system may remain in the oscillatory regime or may settle on a stable steady state. The delays t at which the system crosses a Hopf bifurcation and frequencies y of the oscillations at the bifurcation are determined. The feedback relative to a measured reference species is applied to the inflow of each of the inflow species in turn. For each species used as a delayed feedback, a number of Hopf bifurcations equal to the number of species in the system is located if possible for a system with n species, this would provide bifurcation points with different t and y. As described in detail by Chevalier et al. [7], experimentally determined t and y values can be used to find traces of submatrices of the Jacobian. If n Hopf bifurcations are located for a feedback to each of the species in the system, this allows determination of the complete Jacobian matrix from measurements of a single species in the reaction and then the connectivity of the network can be analyzed. 

Figure 4 Shadowgraph snapshots of the dynamical shape changes in the oscillatory regime. Arrows point in the direction of the main swellingdeswelling dynamics. Snapshots from (a) to (f) at 0, 20, 30, 40, 44, 00 minutes, cover about one period of oscillation of the mid-heigth point of the gel Experimental conditions d=0.5mm, [OH ]o=4-25xlO M. White scale bar in (a)=3mm.

A positive value of the Jacobian (1.19) in the bifurcation point is one of the conditions for oscillatory regime. This is an equivalent of the assertion that the self-oscillations are most probable in the systems where crossed feedbacks have opposite signs. There are some other methods to identify the self-oscillatory systems direct application of Hopf theorem, analysis of type of singular points, Bendixson criterion, reduction of the equation system to Lienard equation, and others. One can find details, for example, in Chap. 4 of [53], or elsewhere [65]. 

Many chemical and biochemical reactions can be in an oscillatory regime in which the concentrations of intermediates and products vary in a regular oscillatory way in time the oscillations may be sinusoidal but usually are not. Sustained oscillations require an open system with a continuous influx of reactants in a closed system oscillations may occur initially when the sjretem is far from equilibrium, but disappear as the system approaches equilibrium. A simple example of an oscillatory reaction is the Selkov model [1]... 

As noted above, for = 0, the system is in the oscillatory regime therefore, for the example shown in Figure 3.12, the oscillations start from the noniUuminated, left end of the gel and the generated wave of swelling and deswelling travels toward... 

The dynamical regime in radio-engineering is self-oscillations. Any real device, such as a neon bold or a vacuum tube, possesses a certain set of adjustable parameters. In practice, the parameter values corresponding to a self-oscillatory regime of the same device, or of a series of similar ones, cannot be exactly identical. Therefore, if a device exhibits repeatedly a similar oscillation, this means that small parameter deviations within some tolerance margins do not change the qualitative character of the process. Naturally, any realistic mathematical model of the system must also exhibit this property of real physical systems. 

In terms of the original variable (p — — ut, the stationary value of (the equilibrium state of system (12.1.9)) corresponds to an oscillatory regime with the same frequency as that of the external force. The periodic oscillations of (the limit cycle in (12.1.9)) correspond to a two-frequency regime. Hence, the above bifurcation scenario of a limit cycle from a homoclinic loop to a saddle-node characterizes the corresponding route from synchronization to beat modulations in Eq. (12.1.7). 

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