Intermittent behavior

The modeling of SMB can be addressed by simulating the system directly, taking into account its intermittent behavior, or by representing its operation in terms of a true countercurrent system. The first model represents the real SMB and considers the periodic switch of the injection and collection points. The second is developed assuming the equivalence with the TMB, where solid and fluid flow in opposite directions. 

Figures 8.8-a through 8.8-d show a few snapshots illustrating intermittent behavior. We have used the logistic-equation driven CML (equation 8.34) and set D = 0.25 and a = 3.83. Each figure shows 16 times overlapped after a certain number of iterations have elapsed (100 iterations in figure 8.7-a, 200 in figure 8,7-b, 300 in 8.7-c, and 500 in 8.8-d). Notice how, depending on the time of the snapshot, some regions are periodic and others are turbulent. Islands cf order tend to come and go as time progresses i.e. the local order is intermittent.

Aging behavior observed in the mean square displacement, (Ax ), as a function of time for different ages. The colloidal system reorganizes slower as it becomes older, (c) y = (Ax )/3 (upper curve) and (Ax ) (lower curve) as a function of the age measured over a fixed time window At = 10 min. For a diffusive dynamics both curves should coincide, however these measurements show deviations from diffusive dynamics as well as intermittent behavior. Panels (a) and (b) from http // www.physics.emory.edu/ weeks/lab/aging.html and Panel (c) from Refill. 

Transformation — Several approaches are available for transformation of time domain data into the - frequency domain, including - Fourier transformation, the maximum entropy method (MEM) [i], and wavelet analysis [ii]. The latter two methods are particularly useful for nonstationary signals whose spectral composition vary over long periods of time or that exhibit transient or intermittent behavior or for time records with unevenly sampled data. In contrast to Fourier transformation which looks for perfect sine... 

One explanation for anomalous diffusion in Hamiltonian dynamics is the presence of self-similar invariant sets or hierarchical structures formed in phase space that play the role of partial barriers. They slow down the normal diffusion. A different explanation for intermittent behavior is given by the existence of deformed and approximate adiabatic invariants in phase space. They are shown in terms of elaborated perturbation theories such as the KAM and Nekhoroshev theorems. 

IIIC) Pomeau, Y., Roux, J. C., Rossi, A., Bachelart, S., Vidal, C. Intermittent Behavior in... 

Figure 9.9. Generation of intermittent behavior through limit point bifurcation in the map /(x) = 0.25 + e + x mod 1 (a) e = 0.55 the system possesses one stable and one unstable fixed point (6) e = 0.02 the fixed points have been destroyed and the system undergoes chaotic bahavior of the intermittent typeP l.

Intermittent behavior is observed away from the jet entrance due to the increase of fluctuations in turbulent jets and shear layers. The high temperature regions correspond to the regions of high mass fraction of combustion products. These regions are located near the stoichiometric surface, where there are ideal conditions of burning. 

Ortoleva, P., Ross, J. (1975) Theory of propagation of discontinuities in kinetic systems with multiple time scales Fronts, front multiplicity, and pulses. J. Chem. Phys. 63, 3398 Pavlidis, T. (1973) Biological Oscillators - Their Mathematical Analysis (Academic, New York) Pomeau, Y., Roux, J. C., Rossi, A., Bachelart, S., Vidal, C. (1981) Intermittent behavior in the Belousov-Zhabotinsky reaction. J. Phys. (Paris) Lett 42, L-271 Rashevsky, N. (1940) An approach to the mathematical biophysics of biological self-regulation and of cell polarity. Bull. Math. Biophys. 2, 15... 

At higher shear rates, an intermittent behavior has been observed, in which the vesicle motion changes in irregular intervals between tumbling and tank-treading [180]. 

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