Oscillation self-sustained oscillations

K. Fichthom, E. Gulari, R. Ziff". Self-sustained oscillations in a heterogeneous catalytic reaction A Monte Carlo simulation. Chem Eng Sci 44 1403-1411, 1989. 

In its applications, science was encountering gradually-increasing difficulties in view of the impossibility of explaining numerous oscillatory phenomena, particularly those connected with the so-called self-sustained oscillations (first, the oscillating arcs and gaseous discharges and still later, the electron tube oscillators). 

The oxidation of CO on Pt is one of the best studied catalytic systems. It proceeds via the reaction of chemisorbed CO and O. Despite its complexities, which include island formation, surface reconstruction and self-sustained oscillations, the reaction is a textbook example of a Langmuir-Hinshelwood mechanism the kinetics of which can be described qualitatively by a LHHW rate expression. This is shown in Figure 2.39 for the unpromoted Pt( 111) surface.112 For low Pco/po2 ratios the rate is first order in CO and negative order in 02, for high pco/po2 ratios the rate becomes negative order in CO and positive order in 02. Thus for low Pcc/po2 ratios the Pt(l 11) surface is covered predominantly by O, at high pco/po2 ratios the Pt surface is predominantly covered by CO. 

D. Durox, T. Schuller, and S. Gandel. Self-sustained oscillations of a premixed impinging jet flame on a plate. Proc. Combust. Inst., 29 69-75, 2002. 

Self-sustained Oscillations. Under certain conditions, isothermal limit cycles in gaseous concentrations over catalysts are observed. These are probably caused by interaction of steps on the surface. Sometimes heat and mass transfer effects intervene, leading to temperature oscillations also. Since this subject has recently been reviewed (42, 43) only a few recent papers will be mentioned here. 

Even though the bifurcation behavior exhibits a Z-shaped curve, it is more complicated due to the existence of the HB. For example, upon ignition, the system is expected to oscillate because no locally stable stationary solutions are found (an oscillatory ignition). Time-dependent simulations confirm the existence of self-sustained oscillations [7, 12]. The envelope of the oscillations (amplitude of H2 mole fraction) is shown in circles (a so-called continuation in periodic orbits). 

The local stability in the neighborhood of the second set of turning points is simply deduced because no new HB point is found the intermediate branch is locally unstable, whereas the partially ignited branch and fully ignited branch are locally stable. The temperature range for self-sustained oscillations is larger at this higher pressure. 

As the pressure increases further, a second HB point (HB2) appears at the extinction point E and shifts toward the other HB HBi) point. An example is shown for 4 atm in Fig. 26.1c. Ignition Ii is no longer oscillatory, because the stationary partially ignited branch becomes locally stable in the vicinity of /i. Time-dependent simulations indicate that the two HB points are supercritical, i.e., self-sustained oscillations die and emerge at these points with zero amplitude. In this case, the first extinction Ei defines again the actual extinction of the system. 

To explain the role of transport, simulations have been also performed in an isothermal PSR. Oscillatory instabilities were again found [8]. These facts indicate that oscillations are radical induced. However, without the heat of reactions, no self-sustained oscillations are found for these conditions. The heat of reactions is a prerequisite at these conditions to pull the HB point outside the multiplic-... 

Figure 26.3 The mole fraction of H2 just above the surface (a) and the wall heat flux (6) as functions of the dimensionless time, 2at, for 10% H2-air mixture at a surface temperature of 1100 K. Self-sustained oscillations and stationary solutions are represented by solid and dashed curves, respectively. The pressure is 4 atm and the strain is a = 200 s ...

Bui, P.-A., D. G. Vlachos, and P. R. Westmoreland. 1997. Self-sustained oscillations in distributed flames modeled with detailed chemistry. Eastern States Section, Chemical and Physical Processes in Combustion Proceedings. Pittsburgh, PA The Combustion Institute. 337-40. 

Neurons of the mammalian suprachiasmatic nucleus (SCN) contain cell-autonomous, self-sustained oscillators, which are able to maintain circadian periodicity even when isolated in vitro or when the animal is placed under... 

Kosbash If you trawl the literature, how many studies have taken a tissue, dissociated it into single cells, prevented the cells from having any kind of contact with each other (including signalling molecules), and then seen longterm self-sustained oscillations Very few, I suspect. 

These studies demonstrate the general mechanism of synchronization of biochemical systems, which I expect to be operative in even more complex systems, such as the mitochondrial respiration or the periodic activity of the slime mold Dictyostelium discoideum. As shown in a number of laboratories under suitable conditions mitochondrial respiration can break into self-sustained oscillations of ATP and ADP, NADH, cytochromes, and oxygen uptake as well as various ion transport and proton transport functions. It is important to note that mitochondrial respiration and oxidative phosphorylation under conditions of oscillations is open for the source, namely, oxygen, as well as with respect to a number of sink reactions producing water, carbon dioxide, and heat. 

Bittger, I., Pettinger, B., Schedel-Niedrig, T. et al. (2001) Self Sustained Oscillations Over Copper in the Catalytic Oxidation of Methanol, Elsevier, Amsterdam. 

Werner, H., Herein, D., Schulz, G. et al. (1997) Catalysis Letters, 49, 109-119. Slin ko, M.M. and Jaeger, N.I. (1994) Physicochemical Basis for the Appearance of Self-Sustained Oscillations in Heterogeneous Catalytic Systems, Elsevier. 

In the resonance region the system oscillates with the external frequency X and with an increased amplitude (entrainment region). Far away from resonance, the internal free oscillations are present. This behaviour is completely absent in systems, where no self-sustained oscillations can exist. A typical example of such a system is a nonlinear conservative system. The resonance diagram has been drawn in Figure 2 for both, the small and the large ampli tude oscillation. 

Before we can start to develop a model we also have to decide how to interpret the behavior observed in Fig. 2.1. The variations in insulin and glucose concentrations could be generated by a damped oscillatory system that was continuously excited by external perturbations (e.g. through interaction with the pulsatile release of other hormones). However, the variations could also represent a disturbed self-sustained oscillation, or they could be an example of deterministic chaos. Here, it is important to realize that, with a sampling period of 10 min over the considered periods of 20-24 h, the number of data points are insufficient for any statistical analysis to distinguish between the possible modes. We need to make a choice and, in the present case, our choice is to consider the insulin-glucose regulation to operate... 

Fig. 2.2 Simulation of a mechanism-based model of ultradian insulin-glucose oscillations. Using independently determined parameters and nonlinear relations, the model displays self-sustained oscillations of the correct period with proper amplitudes and phase relationships. The model also responds correctly to a meal as well as to changes in the rate of glucose infusion.

Figure 2.2 presents the results obtained with our mechanism-based model of the ultradian insulin-glucose oscillations [9], Although clearly only a preliminary model of the phenomenon, the applied model passes all of the above tests. The model produces self-sustained oscillations of the correct period and proper amplitudes, and the model also responds correctly both to a meal and to changes in the rate of glucose infusion. The next step is to use the model to predict the outcome of experiments that have not previously been performed. To the extent that the model is successful in such predictions, the hypothesis underlying the model structure gains additional support. 

With increasing values of S, as we pass the point marked by the black square, the fast subsystem undergoes a Hopf bifurcation. The complex conjugated eigenvalues cross the imaginary axis and attain positive real parts, and the stable focus is transformed into an unstable focus surrounded by a limit cycle. The stationary state, which the system approaches as initial transients die out, is now a self-sustained oscillation. This state represents the spiking behavior. 

The TGF mechanism produces a negative feedback control on the rate of glomerular filtration. However, experiments performed on rats by Leyssac and Baumback [2] and by Leyssac and Holstein-Rathlou [3] in the 1980s demonstrated that the feedback regulation tends to be unstable and to generate large amplitude self-sustained oscillations in the proximal intratubular pressure with a period of 30-40 s. With different amplitudes and phases, similar oscillations have subsequently been observed in the distal intratubular pressure and in the chloride concentration near the terminal part of the loop of Henle [4],... 

For normotensive rats, the typical operation point around a = 10—12 and T = 16 s falls near the Hopf bifurcation point. This agrees with the experimental finding that about 70% of the nephrons perform self-sustained oscillations while the remaining show stable equilibrium behavior [22]. We can also imagine how the system is shifted back and forth across the Hopf bifurcation by variations in the arterial pressure. This explains the characteristic temporal behavior of the nephrons with periods of self-sustained oscillations interrupted by periods of stable equilibrium dynamics. 

Figure 12.6a shows the temporal variation of the proximal tubular pressure Pt as obtained from the single-nephron model for a = 12 and T = 16 s. All other parameters attain their standard values as listed in Table 12.1. Under these conditions the system operates slightly beyond the Hopf bifurcation point, and the depicted pressure variations represent the steady-state limit cycle oscillations reached after the initial transient has died out For physiologically realistic parameter values the model reproduces the observed self-sustained oscillations with characteristic periods of 30-40 s. The amplitudes in the pressure variation also correspond to experimentally observed values. Figure 12.6b shows the phase plot Here, we have displayed the normalized arteriolar radius r against the proximal intratubular pressure. Again, the amplitude in the variations of r appears reasonable. The motion... 

Fig. 12.6 (a) Temporal variation of the proximal tubular pressure Pt as obtained from the single-nephron model fora = 12 and T = 16 s. (b) Corresponding phase plot. With the assumed parameters the model displays self-sustained oscillations in good agreement with the behavior observed for normotensive rats. The unstable equilibrium point falls in the middle of the limit cycle, and the motion along the cycle proceeds in the clockwise direction. 

K.S. Jensen, E. Mosekilde, and N.-H. Holstein-Rathlou, Self Sustained Oscillations and Chaotic Behaviour in Kidney Pressure Regulation, Mondes en Develop. 54/55,91-109 (1986). 

Fig. 36. (A) Self-sustained oscillations during the dissolution of a 3 cm long iron wire in 1 M sulfuric acid at different electrode potentials (a) 0.290 V, (b) 0.285 V, and (c) 0.280 V vs. SCE. (B) Position of the activation front (i.e. during the rising part of the oscillations) versus time. Electrode potentials (a) 0.38 V, (b) 0.34 V, (c) 0.30 V, (d) 0.28 V vs. SCE. (Reproduced from R. Baba, Y. Shiomi, S. Nakabayashi, Chem. Eng. Science 55 (2000) 217 - 222 with permission from Elsevier Science, 2000).

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]. 

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