Soft bifurcation

Experimental observations of regular waves and instability of the main flow lead to search nonlinear steady regimes. The problem of bifurcations of nonlinear steady periodic solutions of (9) was formulated in paper Sh, where the first family of waves was found. This family softly bifurcates from the waveless flow on neutral curve and exists at s G (0,1). Because nonlinear waves of first family move with phase velocity c < 3 they were named as slow waves. The second family of fast waves together with some other bifurcating solutions are obtained by Shkadov et al. (1981), see also Bunov et al. (1984). The full study of intermediate bifurcations is fulfilled as well as full two-parametric manifold of bifurcations is constructed up to now (Sisoev and Shkadov, 1997a, 1999). ... 

These results are thus in agreement with those of bifurcation theory. In the case of odd wave numbers they demonstrate that in general the bifurcation diagrams have to exhibit a subcritical branch. However, there always exists even for odd wave numbers a value of the parameters such that the bifurcation is soft and this value marks the transition from an upper to a lower subcritical branch (see Fig. 21). This feature was less... 

The investigation performed is insufficient for the boundaries of the region of existence of DS to be exactly defined with respect to the parameter i. To define them more exactly, it is necessary that the mode of birth of DS be considered. To this end, we consider different types of the bifurcation diagrams constructed, for example, on the A9 = (9 — min) coordinate, where 9, are a maximum and minimum in the nonuniform temperature profile (see Fig. 11). If, at the bifurcation point the derivative d A9)/dl > 0, the birth is soft (i.e., the uniform mode is replaced by the nonuniform without a jump in amplitude. Fig. 11a). But if d(A9)/dl <0, the birth is hard (i.e., at the critical point, the uniform mode is replaced by the nonuniform when the amplitude reaches some finite value, see Fig. 1 lb). In the second case, as seen from the figure, along with the stable USS there is also a nonuniform state in the length interval l [Pg.569]

Figure 11. Character of bifurcations when a nonuniform solution is born (a) hard birth (b) soft birth (c) hard birth, / - oo.

The bifurcation is of HSAB origin. The iron atom in ferricyanide is quite soft since it is surrounded by sbc cyanide ligands and formal negative charges are present in the anionic cluster. Consequently, the interaction of the iron with the hard oxygen of phenol (phenolate) is unfavorable (symbiotically destabilized). The iron atom of ferric chloride is hard, and the exchange of its ligands with phenols is easy. 

Nunez J and Schmitt P J, Shaped woven tubular soft tissue prostheses and methods of manufacturing , US Patent 5,904,714,1 September 1998. Sidebottom J B, Bifurcated textile fibers and method of weaving the same US Patent 2,845,959,5 August 1958. 

Then, as in the previous case, the sign of the coefficient of A l determines the nature of the bifurcation, direct or inverted (also called soft or hard, respectively). For instance if g = 0 (i o = 0 and buoyancy is set to zero) then Eq. (136) reduces to... 

The first family of 27r-periodic solutions of (18) bifurcates softly from trivial solution /i = 1, g = 1 at neutral value s = 1 and continues into a domain of linear instability s G [0,1]. Let us consider the periodic soluiton including the first two harmonics only... 

Bifurcated rivets. This is a self-piercing rivet, which is used on comparatively soft materials and is shaped like an old-fashioned clothes peg. After the rivet has been driven through the material, the ends are turned over, giving a strong clinch. 

This chapter focuses on the elastic properties of the NEs deduced from shear mechanical measurements. As the comparison between the theoretical descriptions and the experimental results have led to considerable controversies (Martinoty et al. 2004a, b, c Terentjev and Warner 2004 Stenull and Lubensky 2004) it is useful to understand the evolution of the topic in order to make a brief historical presentation of the various static and dynamic theories—conventional linear elastic theory, soft elasticity (original version), soft elasticity (version 2), bifurcation-type theory—which were progressively introduced for describing the elastic properties of these materials. 

The last approach is a non-linear extension of conventional theory, which is called bifurcation-type model. It leads to an effective shear modulus which we will call also C5 in the following for convenience, and which follows the same qualitative behavior as the one predicted by version 2 of soft elasticity (Menzel et al. 2009b). 

Fig 11.5.1. Soft loss of stability of a stable focus at the origin through a supercritical (Li < 0) Andronov-Hopf bifurcation. 

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