Bistability quasi-bistable

The effects of forced oscillations in the partial pressure of a reactant is studied in a simple isothermal, bimolecular surface reaction model in which two vacant sites are required for reaction. The forced oscillations are conducted in a region of parameter space where an autonomous limit cycle is observed, and the response of the system is characterized with the aid of the stroboscopic map where a two-parameter bifurcation diagram for the map is constructed by using the amplitude and frequency of the forcing as bifurcation parameters. The various responses include subharmonic, quasi-peri-odic, and chaotic solutions. In addition, bistability between one or more of these responses has been observed. Bifurcation features of the stroboscopic map for this system include folds in the sides of some resonance horns, period doubling, Hopf bifurcations including hard resonances, homoclinic tangles, and several different codimension-two bifurcations. 

The second part (sections H and I) is devoted to a detailed discussion of the dynamics of unimolecular reactions in the presence and the absence of a potential barrier. Section H presents a critical examination of the Kramers approach. It is stressed that the expressions of the reaction rates in the low-, intermediate-, and high-friction limits are subjected to restrictive conditions, namely, the high barrier case and the quasi-stationary regime. The dynamics related to one-dimensional diffusion in a bistable potential is analyzed, and the exactness of the time dependence of the reaction rate is emphasized. The essential results of the non-Markovian theory extending the Kramers conclusions are also discussed. The final section investigates in detail the time evolution of an unimolecular reaction in the absence of a potential barrier. The formal treatment makes evident a two-time-scale description of the dynamics. 

Let us mention the case of tetrakis(methyltelluro)TTF [76]. This material is (already) in the quasi-ionic phase at 300 K and it undergoes dimerization below 240 K. Some measured values of the threshold field Eth above which jumps of current density occur are = 300 V/cm at 200 K, = 600 V/cm at 180 K, and = 1100 V/cm at 160 K. Crystals have typical dimensions of 0.8 x 0.8 x 0.2 mm3. This material was also chosen to build a prototype bistable device which utilized the switching effect [76]. 

Instead of the quasi-stationary state assumption of Kramers, he assumed only that the density of particles in the vicinity of the top of the barrier was essentially constant. Visscher included in the Foldcer-Planck equation a source term to accoimt for the injection of particles so as to compensate those escaping and evaluated the rate constant in the extreme low-friction limit. Blomberg considered a symmetric, piecewise parabolic bistable potratial and obtained a partial solution of the Fokker-Hanck equation in terms of tabulated functions by requiring this piecewise analytical solution to be continuous, the rate constant is obtained. The result differs from that of Kramers only when the potential has a sharp, nonharmonic barrier. 

As mentioned in Chapter 3, the type of excitable behavior discussed there may be considered as arising from a quasi-bistable dynamics in which one of the involved states, the excited one, is not really stable but lasts only for a finite time. Thus excitable diffusive systems have some similarities with bistable ones, but present an additional level of complexity. 

Apart from the stationary potential patterns just discussed, propagating potential waves under the influence of global (nonlocal) coupling via migration of ions in the electric field are much more readily realized in electrochemical systems [5]. These effects may be most conveniently studied with quasi-one-dimensional systems, that is, ring electrodes where the potential can be recorded at various locations. One example is concerned with the potentiostatic electrochemical oxidation of formic acid on a platinum ring electrode under bistable conditions, as... 

Rotstein, H.G., Zhabotinsky, A.M., Epstein, I.R. Dynamics of one- and two-dimensional kinks in bistable reaction-diffusion equations with quasi-discrete sources of reaction. Chaos 11(4), 833-842 (2001). http //dx.doi.org/10.1063/1.1418459... 

Note that the above mechanism cannot be fulfilled in our experiments, because the induction time of the reaction is quasi-infinite so that the upper limit of the bistable domain, Rg p, takes a quasi-infinite value (10-12), Then it would take a piece of gel of an infinite size to have the reaction switch spontaneously from the unreacted to the reacted state, and the oscillation loop cannot be closed. Work is now in progress in order to devise an experimental system producing oscillations through this mechanism. 

Fig. 12.22 Operation of BiNem bistable device. Two stable states are the uniform (n = 0) and 71-twisted ( = 1). Anchoring is strong at the top plate and weak at the bottom one. A strong field pulse E breaks anchoring and creates a transient quasi-homeotropic texture. If the rear edge of the pulse is short, the backflow develops and the pulse writes a signal in the form of the n-twisted texture. To erase the signal, a strong pulse with a step-like rear edge creates the same transient state, which relaxes to the uniform stable texture due solely to the elastic force...

As seen from Fig. 6.17(c), in the field region < E < Ey, two bistable states exist, leading to homeotropic and focal-conic configuration dependent on the prehistory. The switching between these states takes place via forming the so-called quasi-Grandjean structure at E = E[ with the helix pitch P larger than the equilibrium one P > Pq [62]. The initial focal-conic texture is formed at F7 = 0 [63]. The realization of this thermodynamic bistability depends on the value of d/Po as well as on the liquid crystal elastic anisotropy Kss/Kn, [60, 61]. Let us note that the d/Po... 

Equilibrium dynamics of vesicles comprises the dynamical fluctuations around locally stable mean shapes. Quantitatively, such fluctuations have been studied for quasi-spherical vesicles [29,61] and for prolate shapes in the vicinity of the budding transition [33]. A nontrivial example of dynamical equilibrium fluctuations has been observed at the prolate-oblate transition [62]. As the activation energy between the two locally stable shapes is just a few k T, occasionally thermal fluctuations are large enough to drive the vesicle into the other minimum. This system thus constitutes one of the few examples showing a thermally induced macroscopic bistability. 

Periodic perturbations are also applied to the input rates of acetaldehyde and oxygen simultaneously in the combustion of acetaldehyde in a CSTR. With the two perturbations at the same frequency, we measure bistability and hysteresis as a function of the phase shift between the two perturbations. The application of a perturbation in the flowrate of one reactant to the system already entrained to a perturbation of the flowrate of the second reactant can cause the system to become quasi-periodic in both perturbations to become entrained to both perturbations to remain entrained, but not phase-locked, to the first perturbation or to become quasi-periodic in the first perturbation but entrained to the second perturbation. We measure the effects of frequency-modulated and amplitude-modulated acetaldehyde flowrate perturbations again the results compare well with predictions by a five-variable thermokinetic model.12... 

The use of soluble PDAs in device-like structures is possible because they can be cast to form waveguides or quasi-waveguides. The films have various degrees of order, and this is manifested as a difference in colour the red phase is less ordered than the blue phase. The time-resolved intensity-dependent refractive index n2 for 4-BCMU has been measured by plasmon techniques [110] and found it to be 10" m W" Measurement of interferometric bistability in the same compound [110] has given an appreciably higher figure (5 X 10 m W at 653 and 700 nm). The reason for this discrepancy is not clear, and more work is needed in this area. THG studies on other soluble PDAs also give values around 10 esu, which corresponds to = 10 m [111]. 

The general trends of the scenario are the same as before as D is decreased from infinity, the quasi-homogeneous solution imposed from the boundaries transforms continuously into a two-front pattern. The appearance of these two-front solutions is actually governed by a cusp instability. According to the chosen one parameter path, the transition to the two-front profile solution can be either continuous (as considered in this section) or discontinuous with hysteresis. Note that the experimental observation of spatial bistability... 

Singh and Prasad observed optical bistability in a planar quasi-waveguide at 0.653 Mm and O.7 Mm wavelength with upper switching threshold power of 2.55 MW/cm and 3 86 MW/cm corresponding to extremely large values of nonlinear refractive index 5 10" cm /MW suggesting that it is a thermal effect. 

SINGH B.P. and PRASAD P.N., (1988), 0ptical Bistable Behaviour of a Planar Quasi-Waveguide Interferometer Made with a Conjugated Organic Polymer Film, J. Opt. Soc. Am., B5, 453-456. 

FLC exhibits a natural grayscale with many gray levels, but the FLC switching in this case shows a quasi-bistable behavior with hysteresis, because the helix formation is hindered [11]. 

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