Stochastic oscillator

N. C. Petroni, Z. Marie, Dj. Zivanovic, and J. P. Vigier, Stable states of a relativistic bilocal stochastic oscillator A new quark-lepton model, J. Phys. A (Special Issue Mathematical and General) 14(2), 501-508 (1981). 

E. Scholl, A. Balanov, N. B. Janson, and A. Neiman Controlling stochastic oscillations close to a Hopf bifurcation by time-delayed feedback, Stoch. Dyn. 5, 281 (2005). 

Newell-Whitehead-Segel equation, 23 Non-potential effects, 41 Orientational instability, 283 Pattern formation, 1,11 Phase field, 168 Polymerization wave, 235, 239 Polymerization waves, 236, 238 Propagating front, 260-261 Quantum dots, 123-124 Rayleigh-Benard convection, 61 SHS, 247-248 Smectics, 57 Spiral wave, 47 Stochastic oscillations, 92 Stripes, 2, 10 Surface diffusion, 126... 

I/O data-based prediction model can be obtained in one step from collected past input and output data. However, thiCTe stiU exists a problem to be resolved. This prediction model does not require any stochastic observer to calculate the predicted output over one prediction horiajn. This feature can provide simplicity for control designer but in the pr ence of significant process or measurement noise, it can bring about too noise sensitive controller, i.e., file control input is also suppose to oscillate due to the noise of measursd output... 

Since the first report of oscillation in 1965 (159), a variety of other nonlinear kinetic phenomena have been observed in this reaction, such as bi-stability, bi-rhythmicity, complex oscillations, quasi-periodicity, stochastic resonance, period-adding and period-doubling to chaos. Recently, the details and sub-systems of the PO reaction were surveyed and a critical assessment of earlier experiments was given by Scheeline and co-workers (160). This reaction is beyond the scope of this chapter and therefore, the mechanistic details will not be discussed here. Nevertheless, it is worthwhile to mention that many studies were designed to explore non-linear autoxidation phenomena in less complicated systems with an ultimate goal of understanding the PO reaction better. 

The main features of the copper catalyzed autoxidation of ascorbic acid were summarized in detail in Section III. Recently, Strizhak and coworkers demonstrated that in a continuously stirred tank reactor (CSTR) as well as in a batch reactor, the reaction shows various non-linear phenomena, such as bi-stability, oscillations and stochastic resonance (161). The results from the batch experiments can be suitably illustrated with a two-dimensional parameter diagram shown in Pig. 5. 

A. Boiteux, A. Goldbeter, and B. Hess, Control of oscillating glycolysis of yeast by stochastic, periodic, and steady source of substrate A model and experimental study. Proc. Natl. Acad. Sci. 72(10), 3829 3833 (1975). 

Only deterministic models for cellular rhythms have been discussed so far. Do such models remain valid when the numbers of molecules involved are small, as may occur in cellular conditions Barkai and Leibler [127] stressed that in the presence of small amounts of mRNA or protein molecules, the effect of molecular noise on circadian rhythms may become significant and may compromise the emergence of coherent periodic oscillations. The way to assess the influence of molecular noise on circadian rhythms is to resort to stochastic simulations [127-129]. Stochastic simulations of the models schematized in Fig. 3A,B show that the dynamic behavior predicted by the corresponding deterministic equations remains valid as long as the maximum numbers of mRNA and protein molecules involved in the circadian clock mechanism are of the order of a few tens and hundreds, respectively [128]. In the presence of molecular noise, the trajectory in the phase space transforms into a cloud of points surrounding the deterministic limit cycle. 

Stochastic simulations confirm the existence of bifurcation values of the control parameters bounding a domain in which sustained oscillations occur. The effect of noise diminishes as the number of molecules increases. Only when the maximum numbers of molecules of mRNA and protein become smaller than a few tens does noise begin to obliterate the circadian rhythm. The robustness of circadian rhythms with respect to molecular noise is enhanced when the rate of binding of the repressor molecule to the gene promoter increases [128]. Conditions that enhance the resistance of genetic oscillators to random fluctuations have been investigated [130]. 

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