Multistability

Multistability is the more general notion we start with. It means that the system of differential equations has multiple attractors. Here we only deal with multistationarity, the case in which all the attractors are equilibrium points. 

The last panel in Fig. 3.4 shows a situation qualitatively different from the previous ones. Here, the FN nullclines intersect at three points. Simple linear stability analysis reveals that the central one is unstable, whereas the lateral ones are both stable. Since there are no more attracting structures in this simple dynamical system, the final state of the FN dynamics will be one of these stable fixed points, depending on the initial condition. This situation is called bistability. 

The simplest chemical scheme leading to this behavior is a hypothetical cubic autocatalysis in a CSTR, i.e. one described by the chemical reaction 

This equation always has a stable fixed point at A = Aq, the unreacted state. If k Al 4k0 there are in addition two new fixed points A = (2Aq)-1 (fcii40 (k2Al — 4fcofci) 2)- is linearly stable and linearly unstable. Thus, for k Al 4ko the system is bistable. Note the similarity with Eq. (3.59) as a function of x. The r.h.s. of both are simple cubic polynomials, which when having real roots 

---

Equation A3.11.115(a) is also useful as a fonn that enables easy generalization of the potential scattering theory that we have just derived to multistate problems. In particular, if we imagine that we are interested in the collision of two molecules A and B starting out in states then the asymptotic wavefimction analogous to equation (A3.11.106) is... 

Gray P and Soott S K 1983 Autooatalytio reaotions in the isothermal oontinuous, stirred-tank reaotor isolas and other forms of multistability Chem. Eng. Sc/. 38 29-43... 

Substitution of Eq. (3) into the molecular Schrodinger equation leads to a system of coupled equations in a coupled multistate electronic manifold... 

Abstract. A model of the conformational transitions of the nucleic acid molecule during the water adsorption-desorption cycle is proposed. The nucleic acid-water system is considered as an open system. The model describes the transitions between three main conformations of wet nucleic acid samples A-, B- and unordered forms. The analysis of kinetic equations shows the non-trivial bifurcation behaviour of the system which leads to the multistability. This fact allows one to explain the hysteresis phenomena observed experimentally in the nucleic acid-water system. The problem of self-organization in the nucleic acid-water system is of great importance for revealing physical mechanisms of the functioning of nucleic acids and for many specific practical fields. 

Photochrome flavylium compounds as multistate/multifunction molecular-level systems 99CC107. 

An alternative approach for injection molding is the use of a rotary table injection machine. This type of machine uses the multistation concept, each station having a single cavity mold generally using a point gating/ injection point as previously mentioned. Consider a rotary table machine with eight stations, station one would be the injection station and station seven would be the unload and load metal insert station. This type of process has been successfully used for different types of lip type rotary shaft seals where more expensive elastomers are used and waste needs to be kept to a minimum. 

One color per pass multicolors achieved in multistation units. 

The behavior of the Hg/pc interface is very similar to the Hg/DMF interface 294,301,310 It can be qualitatively described by a multistate model.291 However, although the model can reproduce the electric field and temperature dependencies of the inner-layer properties, the shortcomings of the approach should not be overlooked. 1... 

In principle, one can extract from G(ti)) the complete series of the primary (one-hole, Ih) and excited (shake-up) states of the cation. In practice, one usually restricts the portion of shake-up space to be spanned to the 2h-lp (two-hole, one-particle) states defined by a single-electron transition, neglecting therefore excitations of higher rank (3h-2p, 4h-3p. ..) in the ionized system. In the so-called ADC[3] scheme (22), elertronic correlation effects in the reference ground state are included through third-order. In this scheme, multistate 2h-lp/2h-lp configuration interactions are also accounted for to first-order, whereas the couplings of the Ih and 2h-lp excitation manifolds are of second-order in electronic correlation. 

It has been recognized that the behavior of atomic friction, such as stick-slip, creep, and velocity dependence, can be understood in terms of the energy structure of multistable states and noise activated motion. Noises like thermal activities may cause the atom to jump even before AUq becomes zero, but the time when the atom is activated depends on sliding velocity in such a way that for a given energy barrier, AI/q the probability of activation increases with decreasing velocity. It has been demonstrated [14] that the mechanism of noise activation leads to "the velocity... 

Those with several stations of tooling— multistation (or rotary ) presses... 

These considerations raise a question how can we determine the optimal value of n and the coefficients i < n in (2.54) and (2.56) Clearly, if the expansion is truncated too early, some terms that contribute importantly to Po(AU) will be lost. On the other hand, terms above some threshold carry no information, and, instead, only add statistical noise to the probability distribution. One solution to this problem is to use physical intuition [40]. Perhaps a better approach is that based on the maximum likelihood (ML) method, in which we determine the maximum number of terms supported by the provided information. For the expansion in (2.54), calculating the number of Gaussian functions, their mean values and variances using ML is a standard problem solved in many textbooks on Bayesian inference [43]. For the expansion in (2.56), the ML solution for n and o, also exists, lust like in the case of the multistate Gaussian model, this equation appears to improve the free energy estimates considerably when P0(AU) is a broad function. 

The two expansions discussed so far appear to be quite different. In the multistate Gaussian model, different functions are centered at different values of AU. In the Gram-Charlier expansion, all terms are centered at (AU)0. The difference, however, is smaller that it appears. In fact, one can express a combination of Gaussian functions in the form of (2.56) taking advantage of the addition theorem for Hermite polynomials [44], Similarly, another, previously proposed representation of Pq(AU) as a r function [45] can also be transformed into the more general form of (2.56). 

Hummer, G. Pratt, L. Garcia, A. E., Multistate Gaussian model for electrostatic solvation free energies, J. Am. Chem. Soc. 1997,119, 8523-8527... 

Anonymous (2002) Multistate outbreaks of Salmonella serotype Poona infections associated with eating cantaloupe from Mexico - United States and Canada, 2000-2002 . MMWR Morbidity and Mortality Weekly Reports, 51, 1044—1047. 

Cummings, K., Barrett, E., Mohle-Boetani, J.C., Brooks, J.T., Farrar, J., Hunt, T., Fiore, A., Komatsu, K., Werner, S.B. and Slutsker, L. (2001) A multistate outbreak of Salmonella enterica serotype Baildon associated with domestic raw tomatoes . Emerging Infectious Diseases, 7, 1046-1048. 

Hilborn, E.D., Mermin, J.H., Mshar, P.A., Hadler, J.L., Voetsch, A., Wojtkunski, C., Swartz, M., Mshar, R., Lambert-Fair, M.-A., Farrar, J.A., Glynn, K. and Slutsker, L. (1999) A multistate outbreak of Escherichia coli 0157 H7 infections associated with consumption of mesclun lettuce . Archives of Internal Medicine, 159, 1758-1764. 

The injection-compression process is an important variation in which rubber is injected into a partly closed mould under low pressure used for moulding-on of rubber soles to footwear uppers. Rubber is injected into a gap between the shoe upper and the sole plate, which is held 3-4 mm wider than it would be in its final position. After injection at the first mould station of a multistation machine the sole plate is closed in the manner of compression moulding at the second mould station and rubber is pressed along the sole and over the toe cap. The importance of the process is that it becomes possible to mould without using normal high injection pressures which tear and distort canvas, leather or synthetic uppers. 

The third considered system is described by the potential (x) = 1 — cos (x) — ax. This potential is multistable. We have considered it in the interval [—10,10], taking into account three neighboring minima. We have taken... 

---