Monostable potential

The second considered example is described by the monostable potential of the fourth order (x) = ax4/4. In this nonlinear case the applicability of exponential approximation significantly depends on the location of initial distribution and the noise intensity. Nevertheless, the exponential approximation of time evolution of the mean gives qualitatively correct results and may be used as first estimation in wide range of noise intensity (see Fig. 14, a = 1). Moreover, if we will increase noise intensity further, we will see that the error of our approximation decreases and for kT = 50 we obtain that the exponential approximation and the results of computer simulation coincide (see Fig. 15, plotted in the logarithmic scale, a = 1, xo = 3). From this plot we can conclude that the nonlinear system is linearized by a strong noise, an effect which is qualitatively obvious but which should be investigated further by the analysis of variance and higher cumulants. 

First, we should consider the time evolution of the mean coordinate in the monostable parabolic potential tp(x) = ax1 /2 (linear system), because for this... 

The parameters of the system (35) were chosen such that the potential is monostable ( 2 < 4yce>o), the dependence of the energy of oscillations on their frequency is nonmonotonic (- -7 > and the motion is underdamped... 

Non-periodic oscillations have been observed in the above system in CSTR. The visual oscillations are accompanied by oscillations in the redox potential and I. Oscillations are inhibited by lOj, Cl and NaHC03. These depend on [IO3] [arsenite] and the reciprocal of the residence time. There is an upper limit and a lower limit of [IO3 ] as well as [H3ASO3] in between which oscillations are observed. It is found that these occur in a narrow range of flow rate which is close to the region when the system can switch from the monostable state to a state of bistability [21(b)]. However, for periodic oscillations, different strategies were adopted by Epstein and co-workers [12]. 

The four types of mixed potential models presented in Figs. 12.6 to 12.9 are simplistic and do not necessarily reflect the complete behavior of carbon steel in Kraft liquors because the models all assume some sort of steady states. Figure 12.10 depicts typical curves from an in situ test in a white liquor clarifier at different scan rates. The passive state does not exist until after the active-passive transition is traversed. Therefore, unless sufficient anodic current density is discharged from carbon steel by a naturally occurring cathodic reaction or an applied anodic protection current, the carbon steel liquor interface remains monostable (active) because the passive film and its low current density properties do not exist. 

When K lor K = 1.07 as shown in Fig. 2.5 an interpretation in terms of a society at the crossroads between a liberal and a totalitarian state can be made. The reason for this is the higher willingness of the individual to adapt to the prevailing opinion. As can be seen in Figs. 2.3 b and c and Fig. 2.5 the value iTc = 1 is the critical value of the trend parameter k, where for (5 = 0 the potential (2.129) changes from the monostable to the bistable form when all the possible consequences for bifurcation etc., as discussed in Sect. 2.3, can be observed. 

Group 1 Figures 6.2-4. This group of scenarios is characterized by a relatively weak internal political opinion pressure in both societies [iCi = 1C2 = 0.8 is below the threshold value Kc = 1.0 of a transition from a monostable to a bistable potential Vj (Py,...) see (6.29 a, b)]. Furthermore, because K12 = 2 the political psychology Pi of i is strongly coupled to the corresponding political climate P2 in while, since 1C21 = 0, the reverse is not true. 

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