Homogeneous Oscillations

Fig. 16. Bifurcation diagram of temporal dissipative structures, c (maximal amplitude of the oscillation minus the homogeneous steady-state value) is sketched versus B for a two-dimensional system with zero flux boundary conditions. The first bifurcation occurs at B = Bn and corresponds to a stable homogeneous oscillation. At B, two space-dependent unstable solutions bifurcate simultaneously. They become stable at B a and Bfb. Notice that as it is generally the case Bfa Bfb.

In closed system studies of the BZ reaction, three principal modes of homogeneous oscillations have been identified (1) low-frequency, large amplitude, highly nonlinear (i.e., nonharmonic) relaxation oscillations... 

Fig. 38. Irregular spatiotemporal behavior during H2 oxidation on a Pt ring at a fixed external potential. Electrolyte 0.5 mM H2SO4, 10 5 M Cu2+, 0.1 mM Cl-. (For clarity, the homogeneously oscillating part of the data has been subtracted.)...

A large parameter range in which cluster patterns dominated the spatiotemporal dynamics was also observed for the oscillatory oxidation of H2 in the presence of electrosorbing cations and anions [175], Characteristically, the cluster-type patterns were found at lower concentrations of Cu2+ ions compared with the pulse-type motions shown in Fig. 56. Examples of two-phase and three-phase clusters are depicted in Fig. 63. In these figures the homogeneously oscillating mode has been subtracted. 

Fig. 65. Spatiotemporal evolution of the potential at a Pt electrode during the galvanostatic oxidation of H2 in the presence of Cu2+ and Cl- ions after subtraction of the homogeneous oscillating part [64], (b) Temporal evolution of the total current. The first four oscillations correspond to the time interval shown in (a).

During the recent years a number of studies has been made to design homogeneous oscillators systematically. While the others are the iodate (Bray-Liebhavsky and... 

De Kepper, et al. (1981-1) designed a homogeneous oscillating reaction by coupling the autocatalytic oxidation of arsenite by I03 to the autocatalytic C102 -I03- reaction in a CSTR. Both I2 and I concentrations oscillate with the concentration of the latter changing by a factor of > 105 during each cycle. This arsenite-iodate-chlorite system was obtained in two separate reactions. The oxidation of arsenite by iodate, a reaction autocatalytic in iodide is ... 

Over the past several years there have been many experimental and theoretical studies aimed at developing a better understanding of pattern formation in reaction-diffusion systems. The focus of recent studies has been on more complex behavior away from the onset of instability. For some parameter values, spatiotemporal chaos may occur near the boundary between the Turing region and the region of homogeneous oscillations (Figure 12). 

The time series shows a. stationary pattern until a homogeneously oscillating flow rate around a constant level. However, an additive outlier occurs at time index 128. To analyse the time dependency pattern of the Naphtha time series, the effect of the outlier has to be removed. Therefore, the outlier s effect (i.e. the raise beyond the mean level) is estimated by modelling a simple linear regression model and the corresponding value of the time series is corrected by this estimate. For the corrected Naphtha time series, ACF and PACF are calculated (Figure 2.13a and Figure 2.13b). 

Let me encourage those who have not yet experimented with the Belousov-Zhabotinskii reaction to give it. a try. For your convenience I have given recipes for producing homogeneous oscillations (p. 50) propagating waves (pp. 70f ). The chemicals and glassware are readily available in almost any wet-chemistry laboratory. Just ask ... 

Chemical oscillations were not recognized at early periods of its discovery. Research community did not accept that chemical oscillation is possible and a genuine homogeneous oscillating reaction is impractical topic. Many scientists... 

Kepper, P.D., Kustin, K., Epstein, I.R. A systematically designed homogeneous oscillating reaction the arsenite-iodate-chlorite system. J. Am. Chem. Soc. 103, 2133 (1981)... 

By analysis and numerical simulation, Prigogine and Lefever demonstrated that their model shows homogeneous oscillations and propagating waves like those seen in the BZ system. The Brusselator was extremely important because it showed that a chemically reasonable mechanism could exhibit self-organization. 

This curve is independent of the value of a, and below it the homogeneous steady state is unstable to inhomogeneous perturbation. The Hopf curve, below which the stable homogeneous steady state gives way to homogeneous oscillation, is given by... 

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