Noisy periodicity

So far we have concentrated on the particular parameter values <7 = 10, b =, r = 28, as in Lorenz (1963). What happens if we change the parameters It s like a walk through the jungle—one can find exotic limit cycles tied in knots, pairs of limit cycles linked through each other, intermittent chaos, noisy periodicity, as well as strange attractors (Sparrow 1982, Jackson 1990). You should do some exploring on your own, perhaps starting with some of the exercises. 

Fig. I] (a) noisy periodic signal (b) its PSD (c) reconstruction of the signal using only...

Olsen, L.F. W.M. Schaffer. 1990. Chaos versus noisy periodicity alternative hypotheses for childhood epidemics. Science 249 499-504. 

Figure 12 depicts the time correlation function of the variable x. After a stage of rapid decrease we observe long-living irregular oscillations around zero suggesting that the attractor is organized around a noisy periodic or multi-periodic solution. This is further corroborated by the power spectrum, which features a number of pronounced peaks around a principal frequency w 1.2 and its first few harmonics and subharmonics. 

All of these phenomena arise out of the random reactive and elastic collision events in the system and a fundamental understanding of how macroscopic, self-organized chemical structures appear must be based on descriptions that go beyond the macroscopic, mean-field rate laws or reaction-diffusion equations. In this section we use the reactive lattice-gas method to examine how molecular fluctuations influence oscillatory and chaotic dynamics. In particular we shall show how system size, diffusion, reactions and fluctuations determine the structure of the noisy periodic or chaotic attractors. 

In crystals, the scattering densities are periodic and the Bragg amplitudes are the Fourier components of these periodic distributions. In principle, the scattering density p(r) is given by the inverse Fourier series of the experimental structure factors. Such a series implies an infinite sum on the Miller indices h, k, l. Actually, what is performed is a truncated sum, where the indices are limited to those reflections really measured, and where all the structure factors are noisy, as a result of the uncertainty of the measurement. Given these error bars and the limited set of measured reflections, there exist a very large number of maps compatible with the data. Among those, the truncated Fourier inversion procedure selects one of them the map whose Fourier coefficients are equal to zero for the unmeasured reflections and equal to the exact observed values otherwise. This is certainly an arbitrary choice. 

Wiss [117] and Tumuluri [116] both determined the reaction end point and detected undesired side products or excessive intermediate concentration increases from spectra. Tumuluri went a step further, calculating the reaction kinetics for different catalysts to determine the most efficient and effective one. Interestingly, the reaction with one of the catalysts followed a different mechanism than the others examined. In one reaction, a long induction period was noticed and traced to an unanticipated side reaction. The structure of the side product could not be dehnitively determined because the MCR data reduction was too noisy, but its presence was clearly detected. Additional experiments specihcally designed to study that stage of the... 

If the detector passes the static test, turn on the flow and watch the baseline. Noisy baseline at this point is probably coming from before the detector. Realize that a reciprocating pump is noisy and it now lacks pulse dampening from the column. While we re here, shoot a sample response should be instantaneous, straight up and down. I have not tried it, but you might be able to quantitate lamp strength by shooting a known standard over a time period and measure the deflection. 

The Time Course of Recurrent Mood Disorders Periodic, Noisy and Chaotic Disease Patterns... 

Fig. 7.3 Deterministic (a) and noisy (b) computer simulations of the time course of affective disorders showing the intervals between successive disease episodes (interval duration) as a function of a disease variable S and examples of episode generation from different disease states (figure modified after [2]). In deterministic simulations (a), there is a progression from steady state (S = 18) to subthreshold oscillations (S = 22) with immediate onset of periodic event generation at a certain value of S (slightly below S = 60). With further increase of S, the intervals between successive episodes are continuously...

Kerosene is a good solvent for use with ICP-AES but is prone to noisy plasma. The solvent tetralin (1,2,3,4-tetrahydronaphthalene) has been used by workers involved in metal analysis of crude and lubricating oils with success. The solvent decalin (decahydronaphthalene) was also found to be a good solvent for metal analysis of crude oils but it is very expensive and not used extensively. The analytical performance of these solvents was studied for stability over an extended period of time to determine the effect of varying viscosities. The solvents toluene and xylene are also good solvents for dissolution but have high background to noise ratio and will not be discussed further. 

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