Bistable dynamics

We note that, in the limit in which the Z dynamics is so slow that Z can be considered constant, Zc, there is a range of values of Zc for which the first equation (3.77) is a bistable dynamical system, with coexistence of a low-P and a large-P steady states (May, 1977). 

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. 

Figure 7.9 The time dependence of the total concentration as a function of the rescaled time Da t, at several values of Da for the bistable dynamics in the closed flow Da increases from top to bottom, dashed line is the homogeneous result, which is approached as Da —> 0. The inset plots the evolution in terms of the unsealed time. Dashed line in the inset gives the exponential length growth of a material line in the same flow.

Figure 7.10 Total average concentration Ctotai in the stationary state vs Da obtained numerically for the bistable model in the open vortex-sink flow. Note the discontinuous jump to Ctotai = 0 at Da = Dac R 24.2, that is characteristic to the bistable dynamics.

We have presented a method to calculate the mean frequency and effective diffusion coefficient of the numbers of cycles(events) in periodically driven renewal processes. Based on these two quantities one can evaluate the number of locked cycles in order to quantify stochastic synchronization. Applied to a discrete model of bistable dynamics the theory can be evaluated analytically. The system shows only 1 1 synchronization, however in contrast to spectral based stochastic resonance measures the mean number of locked cycles has a maximum at an optimal driving frequency, i.e. the system shows bona fide resonance [6]. For the discrete model of... 

C. Flamet, E. Clement, P. Leroux-Hugon, L. M. Sander. Exact dynamics of a bistable chemical reaction model. J Phys A (Math Gen) 25 L1317-L1322, 1992. 

Fig. 6. Dynamical phase diagram of the ascorbic acid/copper(II)/oxygen system in a CSTR in the kf — [Cu2+]0 plane. Fixed reactor concentrations [H2Asc]0 = 5.0x10 4M [H2SO4]0 = 6.0 x 10-5 M [Na2SO4]0 = 0.04M. Symbols O, steady state , oscillations , bistability. The asterisk ( ) marks the Takens-Bogdanov point. Strizhak, P. E. Basylchuk, A. B. Demjanchyk, I. Fecher, F. Shcneider, F. W. Munster, A. F. Phys. Chem. Chem. Phys. 2000, 2, 4721. Reproduced by permission of The Royal Society of Chemistry on behalf of the PCCP Owner Societies.

Interestingly, the simple model is already sufficient to exhibit a variety of dynamic regimes, including bistability and oscillations. 

The interplay between oscillations and bistability has been addressed in detailed molecular models for the cell cycles of amphibian embryos, yeast and somatic cells [138-141]. The predictions of a detailed model for the cell cycle in yeast were successfully compared with observations of more than a hundred mutants [142]. Other theoretical studies focus on the dynamical properties of particular modules of the cell cycle machinery such as that controlhng the Gl/S transition [143]. 

In order to make the copper central core as easy as possible to rotate, we thought that the bulky stoppers should be located far away from the central complex. We thus prepared and studied a new bistable rotaxane, depicted in Fig. 14.10, whose stoppers are indeed very remote from the copper center.31 This new dynamic system can indeed be set in motion more rapidly than the previously described systems. 

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. 

The formaldehyde-sulfite reaction displays non-linear dynamics it is a clock reaction with a sudden pH excursion (from ca 7 up to ll).280 The induction period in batch processes is explained by the internal buffer systems, HS03 -S03. However, flow reactors also exhibit pH oscillations and bistability. 

Flexibility recent reports on the dynamic properties of PCPs show that they are much more flexible than generally believed. Dynamic pores can form a type of soft framework with bistability, whose two states oscillate back and forth between one of two counterparts. A system can exist in either of two states for different parameters of an external field. The structural rearrangement... 

In the following the dynamic properties of N-NDR and S-NDR systems are compiled. We start out by reviewing causes for the occurrence of an N-shaped current-potential curve. Subsequently, the conditions for bistability in one-variable N-NDR systems are discussed. Then, two classes of N-NDR oscillators with dynamically different behaviors are introduced and finally the properties of S-NDR systems are summarized. For both types of NDRs, the most important representatives of the classes are reviewed, whereby the very recent examples that are not discussed in earlier review-type articles as well as those systems with which most of the spatial measurements were done are in the foreground. 

The existence of bistability in the //under conditions under which chemical variable, on which the current depends, exhibits bistability as a function of DL. Thus, in S-NDR systems we have to require that the dynamic equations contain a chemical autocatalysis. As set forth below, m takes the role of the negative feedback variable. The positive feedback might be due to chemical autocatalytic reaction steps as is the case in Zn deposition [157, 158] or CO bulk oxidation on Pt [159], S-shaped current-potential characteristics may also arise in systems with potential-dependent surface phase transitions between a disordered (dilute) and an ordered (condensed) adsorption state due to attractive interactions among the adsorbed molecules. 

Eq. (42) gives rise to a negative feedback loop if the current potential curve is S-shaped, but not for Z-shaped characteristics. Thus, in S-NDR systems DL may stabilize the middle branch of the S, or it may induce oscillations. This is not possible in Z-shaped systems, where an incorporation of DL in the dynamic description only increases the width of the bistable region but never results in qualitatively different behavior. For this reason, DL is not an essential variable in the latter type of systems. Thus, they have to be classified as systems with chemical instabilities only and will not be further treated here. 

Fig. 32. Illustration of front motion in a bistable system due to the interplay of homogeneous dynamics and migration coupling (see text).

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