Oscillatory system

Another important reaction supporting nonlinear behaviour is the so-called FIS system, which involves a modification of the iodate-sulfite (Landolt) system by addition of ferrocyanide ion. The Landolt system alone supports bistability in a CSTR the addition of an extra feedback chaimel leads to an oscillatory system in a flow reactor. (This is a general and powerfiil technique, exploiting a feature known as the cross-shaped diagram , that has led to the design of the majority of known solution-phase oscillatory systems in flow... 

Note that this latter method differs from the midpoint method, where one would use r(q +i/2) = (qn+i + qn)/2 instead of (7c) for r +i/2 in (7b). For highly oscillatory systems with k e, this can be a significant difference, because r is discretized directly in (7). An example in 4 below shows that the midpoint method can become unstable while (7) and (6) remain. stable. 

Y. A. Mitropolsky, Nan-stationary processes in nonlinear oscillatory systems, English translation by Air Technical Intelligence Center, Ohio. 

In general, parametric excitation (or action) may be defined as follows if a parameter of an oscillatory system is made to vary periodically with frequency 2/, / being the free frequency of the system, the latter begins to oscillate with its own frequency. 

Leopold et al. and Nyholm et al. have investigated this oscillatory system by in situ confocal Raman spectroscopy [43], and in situ electrochemical quartz crystal microbalance [44], and in situ pH measurement [45] with the focus being on darification of the osdllation mechanism. Based on the experimental results, a mechanism for the oscillations was proposed, in which variations in local pH close to the electrode surface play an essential role. Cu is deposited at the lower potentials ofthe oscillation followed by a simultaneous increase in pH close to the surface due to the protonation... 

If we assume that an oscillatory system response can be fitted to a second order underdamped function. With Eq. (3-29), we can calculate that with a decay ratio of 0.25, the damping ratio f is 0.215, and the maximum percent overshoot is 50%, which is not insignificant. (These values came from Revew Problem 4 back in Chapter 5.)... 

Iwasaki H, Dunlap JC 2000 Microbial circadian oscillatory systems in Neurospora and Synechococcus models for cellular clocks. Curr Opin Microbiol 3 189-196 Lee K, Loros JJ, Dunlap JC 2000 Interconnected feedback loops in the Neurospora circadian system. Science 289 107-110... 

One particular pattern of behaviour which can be shown by systems far from equilibrium and with which we will be much concerned is that of oscillations. Some preliminary comments about the thermodynamics of oscillatory processes can be made and are particularly important. In closed systems, the only concentrations which vary in an oscillatory way are those of the intermediates there is generally a monotonic decrease in reactant concentrations and a monotonic, but not necessarily smooth, increase in those of the products. The free energy even of oscillatory systems decreases continuously during the course of the reaction AG does not oscillate. Nor are there specific individual reactions which proceed forwards at some stages and backwards at others in fact our simplest models will comprise reactions in which the reverse reactions are neglected completely. 

There is a period of time or, if we prefer, a range of reactant concentration over which the system spontaneously moves away from the pseudo-stationary state. The idea that stationary states may be unstable is not widely appreciated in chemical kinetics but it is fundamental to the analysis of oscillatory systems. 

In 1977. Professor Ilya Prigogine of the Free University of Brussels. Belgium, was awarded Ihe Nobel Prize in chemistry for his central role in the advances made in irreversible thermodynamics over the last ihrec decades. Prigogine and his associates investigated Ihe properties of systems far from equilibrium where a variety of phenomena exist that are not possible near or al equilibrium. These include chemical systems with multiple stationary states, chemical hysteresis, nucleation processes which give rise to transitions between multiple stationary states, oscillatory systems, the formation of stable and oscillatory macroscopic spatial structures, chemical waves, and Lhe critical behavior of fluctuations. As pointed out by I. Procaccia and J. Ross (Science. 198, 716—717, 1977). the central question concerns Ihe conditions of instability of the thermodynamic branch. The theory of stability of ordinary differential equations is well established. The problem that confronted Prigogine and his collaborators was to develop a thermodynamic theory of stability that spans the whole range of equilibrium and nonequilibrium phenomena. 

Finally, the quest to develop mechanistic explanations for these varied and fascinating phenomena can succeed only if more data become available on the component processes. Kinetics studies of the reactions which make up a complex oscillatory system are essential to its understanding. In some cases, traditional techniques may be adequate, though in many others, fast reaction methods will be required. There also appears to be some promise in developing an analysis of the relaxation of flow systems in non-equilibrium steady states as a technique to complement equilibrium relaxation techniques. 

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... 

Derived from Hodgkin-Huxley s celebrated theory and inspired by the experimental observations, cellular calcium dynamics, either stimulated via inositol 1,4,5-trisphosphate (IP3) receptor in many non-muscle cells [69,139], or via the ryanodine receptor in muscle cells [108], is another extensively studied oscillatory system. Both receptors are themselves Ca2+ channels, and both can be activated by Ca2+, leading to calcium-induced calcium release from endoplasmic reticulum. 

Example 13.2 Van der Pol s equations Van der Pol s equations provide a valuable framework for studying the important features of oscillatory systems. It describes self-sustaining oscillations in which eneigy is fed into small oscillations and removed from large oscillations. Consider the following system of ordinary differential equations called Van der Pol s equations... 

Van der Pol s equations provide a valuable framework for studying the important features of oscillatory systems. 

The main problem in the solution of non-linear ordinary and partial differential equations in combustion is the calculation of their trajectories at long times. By long times we mean reaction times greater than the time-scales of intermediate species. This problem is especially difficult for partial differential equations (pdes) since they involve solving many dimensional sets of equations. However, for dissipative systems, which include most applications in combustion, the long-time behaviour can be described by a finite dimensional attractor of lower dimension than the full composition space. All trajectories eventually tend to such an attractor which could be a simple equilibrium point, a limit cycle for oscillatory systems or even a chaotic attractor. The attractor need not be smooth (e.g., a fractal attractor in a chaotic system) and is in some cases difficult to compute. However, the attractor is contained in a low-dimensional, invariant, smooth manifold called the inertial manifold M which locally attracts all trajectories exponentially and is easier to find [134,135]. It is this manifold that we wish to investigate since the dynamics of the original system, when restricted to the manifold, reduce to a lower dimensional set of equations (the inertial form). The inertial manifold is, therefore, a useful notion in the field of mechanism reduction. 

In oscillatory systems, due to the nonlinearity and the variations in parameters multiple solutions, both oscillatory and nonoscillatory simultaneously may appear. Multiple stationary solutions have been discussed by Bilous and Amundson (1955), Aris and Amundson (1958), see Section III.D. Furthermore, multiple oscillating solutions have been extensively studied by Sel kov and his collaborators, see Section III.F. 

III H) 1910-1 Lotka, A. Contribution to the Theory of Periodic Reactions, J. Phys. Chem. vol. 14, (III C) 1975 Marek, M., Svobodova, E. Nonlinear Phenomena in Oscillatory Systems of Homogeneous Reactions-Experimental Observations, Biophysical Chemistry, vol. 3, 263—273 (III A) 1974 Matsuzaki, I., Nakajima, T., Liebhafsky, H. A. The Mechanism of the Oscillatory Decomposition of Hydrogen Peroxide by the I2-IO3-couple, Chem. Letters (Japan) 1463-1466 (IIT O) 1969 Nakamura, S., Yokota, K., Yamazaki, I. Sustained Oscillations in a Lactoperoxidase, NADPH and 02 system Nature vol. 222, 794... 

Although this reaction is the oldest among the oscillatory reactions known to chemists, it has not yet been fully explored. However, a complete understanding of this oscillatory systems might possibly shed light on iodide-peroxide interactions in biological systems. We group the recent studies into two subsections. 

IIIC) Burger, M., Koros, E. Chemistry of the Belousov-Zhabotinskii Oscillatory Systems. 

---