Semi-stable curve

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... 

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. 

A mathematical analysis capable of handling these situations may be slower to emerge than one which treats only the isolated space curve with twist, in a homogeneous unbounded medium. Our numerical experiments are intended to encourage the latter development by providing both suggestive empirical rules for filament dynamics, and tests of competitive semi-analytical approximations. We may at the same time discover completely stable organizing centers whose observed symmetries may lead to a simpler theory. 

Remark. In the multi-dimensional case where besides the central coordinates there are also the stable ones, the unstable set consists of three curves, whereas the stable set is a bimch consisting of three semi-planes intersecting along the non-leading manifold as shown in Fig. 10.5.4, for the three-dimensional example. 

Later, a more detailed investigation of the evolution of hydrogen at tungsten[177] showed that the shape of the polarization curve described above, i.e. the existence of two Tafel regions, is unstable and corresponds to a surface on which an indefinite quantity of oxide is retained. A cleaner surface, on which stable values of overpotential can be obtained almost without any hysteresis for forward and backward runs, is characterized by the absence of straight semi-logarithmic sections and by a continuous increase in the slope. In acidic and alkaline solutions, the overpotential is found to be almost independent of pH both in salts and in pure alkalis. Besides, it is also found to be independent of the nature of the cation (Li" ", Na, K", Cs" ", Ba, N(Ct H9)J). The electrode has a large adsorption capacity. All these facts indicate the predominance of a slow recombination mechanism. 

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