Nonlinear region stability conditions

The condition (10) allows one to determine regions in the q, p)-plane corresponding to different types of pattern excitation and stability near threshold as shown in Fig.3. The straight hnes OCB and OGF correspond to Ao = 0 and the curves AB, CD, EF, GH are parts of the hyperbola sw/m + Xq = 0. It is interesting that at the intersection point O, p = 1/2, q = 3/4. Since p = 1/2 corresponds to c = 0, this means that unless the wetting potential depends on the film slope the periodic structure is always subcritical and therefore blows up. Weakly nonlinear analysis is not useful in this case. 

A front corresponds to a traveling wave solution, which maintains its shape, travels with a constant velocity v, p x, t) = p(x - v t), and joins two steady states of the system. The latter are uniform stationary states, p(x, t) = p, where Ffp) = 0. For the logistic kinetics, the steady states are = 0 and jo2 = 1- While the logistic kinetics has only two steady states, three or more stationary states can exist for a broad class of systems in nonlinear chemistry and population dynamics with Alice effect, but a front can only connect two of them. To determine the propagation direction of the front, we need to evaluate the stability of the stationary states, see Sect. 1.2. The steady state jo is stable if P (fp) < 0 and unstable if F (jo) > 0. Let the initial particle density p x,0) be such that on a certain finite interval, p x,0) is different from 0 and 1, and to the left of this interval p(x,0) = 1, while to the right p x, 0) = 0. In this case, the initial condition is said to have compact support. Kolmogorov et al. [232] showed for Fisher s equation that due to the combined effects of diffusion and reaction, the region of density close to 1 expands to the... 

Physically, the hysteresis roots in that fact that the effect of the electric force on the stability of 1D conduction is different in different parts of the diffusion layer. Indeed, this force stabilizes ID conduction in the electroneutral bulk and in the quasi-equUibrium portion of EDL and destabilizes it in the ESC region. The nonlinear flow resulting from this instability reduces concentration polarization and, thus, weakens the hampering effect of the electric force in the bulk in the down way portion of the hysteresis loop. In order to verify this mechanism, a model electroosmotic formulation without electric force term in the Stokes equation was analyzed. As illustrated in Fig. 8, this modification results in shrinking of the hysteresis loop. The bifurcation still remains subcritical and the hysteresis loop still exists owing to the hampering effect of the electric force in the quasi-equilibrium portion of the EDL, implicit in the first term in the electroosmotic slip conditions (21). 

In this section, we discuss in detail how the selection of various experimental parameters affects each of these conditions. One of the first studies on EIS measurements in MXC applications by Strik et al. [29] covers some of these conditions very well, but we provide an expanded explanation here. While all these conditions are especially difficult to fulfill in a typical electrochemical cell, the conditions used in MXCs further exacerbate the problem. For example, it is known that polarization curves for microbial anodes exhibit nonlinear, Nernstian responses [30]. Thus, there are regions in the polarization curve where the system may not behave linearly even when small amplitudes are applied. The irreversibility of the enzymatic responses also leads to regions where finiteness is not met (Fig. 8.6). These cases would run also into difficulties in terms of the signal-to-noise ratio when small amplitudes are applied at potentials on the saturation region of the polarization curve. Similarly, as MXCs are biological reactors and can have changes in microbial responses due to small perturbations outside the conttol of researchers, conditions of both stability and causality are difficult to fulfill. 

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