Dynamic oscillatory

Recently, Vigil and Willmore [67] have reported mean field and lattice gas studies of the oscillatory dynamics of a variant of the ZGB model. In this example oscillations are also introduced, allowing the reversible adsorption of inert species. Furthermore, Sander and Ghaisas [69] have very recently reported simulations for the oxidation of CO on Pt in the presence of two forms of oxygen, namely chemisorbed atomic O and oxidized metal surface. These species, which are expected to be present for reaction under atmospheric pressure, are relevant for the onset of oscillatory behavior [69]. 

R. D. Vigil, F. T. Willmore. Oscillatory dynamics in a heterogeneous surface reaction Breakdown of the mean-field approximation. Phys Rev E 54 1225-1231, 1996. 

Once the parametric representation of the Jacobian is obtained, the possible dynamics of the system can be evaluated. As detailed in Sections VILA and VII.B, the Jacobian matrix and its associated eigenvalues define the response of the system to (small) perturbations, possible transitions to instability, as well as the existence of (at least transient) oscillatory dynamics. Moreover, by taking bifurcations of higher codimension into account, the existence of complex dynamics can be predicted. See Refs. [293, 299] for a more detailed discussion. 

From a mathematical point of view, the onset of sustained oscillations generally corresponds to the passage through a Hopf bifurcation point [19] For a critical value of a control parameter, the steady state becomes unstable as a focus. Before the bifurcation point, the system displays damped oscillations and eventually reaches the steady state, which is a stable focus. Beyond the bifurcation point, a stable solution arises in the form of a small-amplitude limit cycle surrounding the unstable steady state [15, 17]. By reason of their stability or regularity, most biological rhythms correspond to oscillations of the limit cycle type rather than to Lotka-Volterra oscillations. Such is the case for the periodic phenomena in biochemical and cellular systems discussed in this chapter. The phase plane analysis of two-variable models indicates that the oscillatory dynamics of neurons also corresponds to the evolution toward a limit cycle [20]. A similar evolution is predicted [21] by models for predator-prey interactions in ecology. 

Bui, P.-A., D. G. Vlachos, and P. R. Westmoreland. 1999. On the local stability of multiple solutions and oscillatory dynamics of spatially distributed flames. Combustion Flame 117 307-22. 

In a series of experiments we have tested the type and range of entrainment of glycolytic oscillations by a periodic source of substrate realizing domains of entrainment by the fundamental frequency, one-half harmonic and one-third harmonic of a sinusoidal source of substrate. Furthermore, random variation of the substrate input was found to yield sustained oscillations of stable period. The demonstration of the subharmonic entrainment adds to the proof of the nonlinear nature of the glycolytic oscillator, since this behavior is not observed in linear systems. A comparison between the experimental results and computer simulations furthermore showed that the oscillatory dynamics of the glycolytic system can be described by the phosphofructokinase model. 

The attenuation of die dipole of the repeat unit owing to thermal oscillations was modeled by treating the dipole moment as a simple harmonic oscillator tied to the motion of the repeat unit and characterized by the excitation of a single lattice mode, the mode, which describes the in-phase rotation of the repeat unit as a whole about the chain axis. This mode was shown to capture accurately the oscillatory dynamics of the net dipole moment itself, by comparison with short molecular dynamics simulations. The average amplitude is determined from the frequency of this single mode, which comes directly out of the CLD calculation ... 

Finally, Section 2.4 analyses a simplified model of a bursting pancreatic /3-cell [12]. The purpose of this section is to underline the importance of complex nonlinear dynamic phenomena in biomedical systems. Living systems operate under far-from-equilibrium conditions. This implies that, contrary to the conventional assumption of homeostasis, many regulatory mechanisms are actually unstable and produce self-sustained oscillatory dynamics. The electrophysiological processes of the pancreatic /3-cell display (at least) two interacting oscillatory processes A fast process associated with the K+ dynamics and a much slower process associated with the Ca2+ dynamics. Together these two processes can explain the characteristic bursting dynamics in the membrane potential. 

We have shown computer simulations from all these different levels First, we modeled the time course of affective disorders and showed that clinical observation can be mimicked in remarkable details with a combination of oscillatory dynamics and noise. Second, we presented an initial, still very basic, model of circadian cortisol release which nevertheless provided new insights also into eventual disease relevant alterations. Third, we showed single neuron and neuronal network simulations to elucidate the relevant interdependencies between ionic conductances and network interactions with regard to neuronal synchronization at different dynamic, also heterogeneous states. 

IIIN) Heinemann, R. F., Poore, A. B. Multiplicity Stability, and Oscillatory Dynamics of the... 

The region of oscillatory dynamics further reduces when the damping of vibration modes is taken into account. An analysis based on Hurwitz criterion reveals that exponential amplification takes place for almost all values of the parameters except for a class of special configurations meeting [127]... 

Chemical reactions in the atmosphere form a large system of interacting chemical species described by nonlinear kinetics. It has been shown that certain components of this system can also exhibit oscillatory dynamics in some range of the parameters (see e.g. Poppe and Lustfeld (1996)). When the typical period of these oscillations... 

When the stirring rate is below a certain threshold the synchronized oscillation disappears and the mean concentration is almost constant apart from small irregular fluctuations that are independent of the stirring rate. In this regime the mixing is too slow to strongly couple the oscillatory dynamics of the fluid parcels. Snapshots of the... 

The autocatalytic biochemical model with recycling of product suggests that birhythmicity could occur in neurons of both the inferior olive and the thalamus. A detailed appraisal of the role played by the oscillatory dynamics of the olivary neurons in the coordination of motor control has recently been given by Welsh et al. (1995). 

Doedel, E.J. R.F. Heinemann. 1983. Numerical computation of periodic solution branches and oscillatory dynamics of the stirred tank reactor with A —> B C reactions. Chem. Eng. Sci. 38 1493-9. 

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