Semi-stable cycle

Fig- 11.3.5. Bifurcations of a saddle-node limit cycle in a plane for the case /2 > 0- A semi-stable cycle (b) is attractive in the exterior domain, but repelling in the interior region. 

Fig. 11.4.7. The map for the case h when moving counter-clock-wise in the direction around the origin in the bifurcation diagram in Fig. 11.4.6(a). Two period-two cycles in (c) coalesce on the border separating D2 a-nd Do and disappear in Do. The semi-stable cycle of period two is shown in (d).

Theorem 11.4 shows essentially that outside the narrow sector bounded by 1 and 2, the bifurcation behavior does not differ from that of equilibrium states (see Sec. 11.5) fixed points correspond to equilibrium states, and the invariant curves correspond to periodic orbits. However, the transition from the region D2 to the region Dq occurs here in a more complicated way. In the case of equilibrium states the regions D2 and Do are separated by a line on which a stable and an imstable periodic orbits coalesce thereby forming a semi-stable cycle. In the case of invariant closed curves, the existence of a line corresponding to a semi-stable invariant closed curve is possible only in very degenerate cases (for example, when the value of R does not depend on as... 

The catalytic cycle of peroxidases (Fig. 5) begins with the oxidation of the high-spin, pentacoordinate ferric native enzyme 10) by hydrogen peroxide to form a semi-stable intermediate called Compound I (//). Compound I is a high-valent oxo-iron complex that is two oxidation equivalents above ferric horseradish peroxidase. Although formally an Fe heme. Compound 1 is generally thought to be an Fe porphyrin 71-cation radical [51, 52]. 

As for all basic bricks, refractoriness of dolomite is also very high—above 1750°C. The RUL is 1450°C to 1550°C for stabilized dolomite and 1350°C to 1450°C for semi-stable dolomite. The spalling resistance of stabilized dolomite is poor, but the resistance of semi-stable dolomite is moderately good. Semi-stable dolomite can withstand 20 cycles. Slag resistance is poor for both the types of dolomite. In order to increase this resistance, dolomite is doped with magnesia, which combines with silica to form forsterite. Forsterite has better resistance to slag. Semi-stable dolomite has better resistance than stabilized dolomite. 

If the Lyapunov value h 0 here, then the associated double fixed point corresponds to a double (semi-stable) limit cycle of the original system. 

D) if there is a semi-stable (double) limit cycle, the system may not have simultaneously an unstable separatrix of a saddle which tends to the cycle as t -> -hoo and a stable separatrix of a saddle which tends to the cycle as t —00, as shown in Fig. 8.1.4 and... 

This bifurcation diagram for the equilibrium state with two zero characteristic exponents had been known for a long time. However, there remained a problem of proving the uniqueness of the limit cycle. In other words, one must prove additionally that there are no other bifurcational curves besides Li,...,L4 (namely, curves corresponding to semi-stable limit cycles). This problem was independently solved by Bogdanov [33] and Takens [146] with whom this bifurcation is often named after. 

The above process was observed only in the initial cycles. Nevertheless, any electrochemical reduction of As(V) would raise concern about the safety of using LiAsFe in a commercial battery, because, while arsenate in its high oxidation state (V) is not particularly toxic, As(III) and As(0) species From the electrochemical point of view, however, the above reduction could be a benefit, especially for lithium ion cells, since an SET formed on an anode at > 1.0 V vs lithium would be very stable during the operation of a lithium ion cell according to a semi-empirical rule, ° which will be discussed in more detail in section 6. 

Recent advances on the Ca-Br cycle were presented in an ANL paper. The original concept for this cycle involved solid phase reactions in a semi-continuous batch operation. The ANL paper reported on experiments that used a direct sparging reactor in the hydrolysis reaction to allow continuous production of HBr which is then electrolytically decomposed to produce hydrogen. The sparging steam was introduced into the molten bath of CaBr2 which yielded HBr in a stable and continuous operation. 

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