Bifurcation types

Fig. 3 Parallel lamination mlcromlxer types (a) Bifurcation-type feeds (Adapted from [66] with kind permission from Springer Science), (b) Interdigital-type feeds, super focus mixer (Reproduced from [67] with permission. Copyright Wiley-VCH ). (c) Chessboard micromixer (Adapted from [68] with permission. Copyright lOP Publishing), (d) Circular micromixer (Adapted from [69] with permission. Copyright lOP Publishing)...

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). 

Figure C3.6.6 The figure shows tire coordinate, for < 0, of tire family of trajectories intersecting tire Poincare surface at cq = 8.5 as a function of bifurcation parameter k 2- As tire ordinate k 2 decreases, tire first subhannonic cascade is visible between k 2 0.1, tire value of tire first subhannonic bifurcation to k 2 0.083, tire subhannonic limit of tire first cascade. Periodic orbits tliat arise by tire tangent bifurcation mechanism associated witli type-I intennittency (see tire text for references) can also be seen for values of k 2 smaller tlian tliis subhannonic limit. The left side of tire figure ends at k 2 = 0.072, tire value corresponding to tire chaotic attractor shown in figure C3.6.1(a). Otlier regions of chaos can also be seen.

One may also observe a transition to a type of defect-mediated turbulence in this Turing system (see figure C3.6.12 (b). Here the defects divide the system into domains of spots and stripes. The defects move erratically and lead to a turbulent state characterized by exponential decay of correlations [59]. Turing bifurcations can interact with the Hopf bifurcations discussed above to give rise to very complicated spatio-temporal patterns [63, 64]. 

Bifurcation buckling occurs at the load at which the load path forks into two load paths, the new one stable and the other, the continuation of the old path, unstable, irrespective of the type of deformation path prior to buckling (linear or nonlinear). 

Weak intramolecular interactions between sulfur or selenium and nitrogen are a recurrent phenomenon in large biomolecules. They may occur in the same residue or between neighbours of a peptide chain. The formation of four- or five-membered rings of the types 15.1 and 15.2, respectively, is most common. A feature that is unique to proteins is the participation of sulfur atoms in bifurcated N S N contacts. 

There are two principal types of bifurcation phenomena those of the first kind and those of the second kind. ... 

In the preceding sections, various types of fluctuations and instabilities essential to corrosion were examined. As a result, it was shown that a corrosion system involves various kinds of problems of stability and instability. Unlike thermodynamic equilibrium systems, in nonequilibrium systems like corrosion systems, a drastic change in the reaction state should be defined as a bifurcation phenomenon. 

There is another type of bifurcation called Turing bifurcation, which results in a spatial pattern rather than oscillation. A typical example where a new spatial structure emerges from a spatially unique situation is Benard s convection cells. These have been well examined and are formed with increasing heat conduction.53 Prigogine called this type of structure a dissipative structure.54-56... 

What cannot be obtained through local bifurcation analysis however, is that both sides of the one-dimensional unstable manifold of a saddle-type unstable bimodal standing wave connect with the 7C-shift of the standing wave vice versa. This explains the pulsating wave it winds around a homoclinic loop consisting of the bimodal unstable standing waves and their one-dimensional unstable manifolds that connect them with each other. It is remarkable that this connection is a persistent homoclinic loop i.e. it exists for an entire interval in parameter space (131. It is possible to show that such a loop exists, based on the... 

The lower a graph is more interesting. While initially the Poincar6 phase portrait looks the same as before (point E, inset 2c) an interval of hysteresis is observed. The saddle-node bifurcation of the pericxiic solutions occurs off the invariant circle, and a region of two distinct attractors ensues a stable, quasiperiodic one and a stable periodic one (Point F, inset 2d). The boundary of the basins of attraction of these two attractors is the one-dimensional (for the map) stable manifold of the saddle-type periodic solutions, SA and SB. One side of the unstable manifold will get attract to one attractor (SC to the invariant circle) while the other side will approach die other attractor (SD to die periodic solution). 

The mechanism of these transitions is nontrivial and has been discussed in detail elsewhere Q, 12) it involves the development of a homoclinic tangencv and subsequently of a homoclinic tangle between the stable and unstable manifolds of the saddle-type periodic solution S. This tangle is accompanied by nontrivial dynamics (chaotic transients, large multiplicity of solutions etc.). It is impossible to locate and analyze these phenomena without computing the unstable, saddle-tvpe periodic frequency locked solution as well as its stable and unstable manifolds. It is precisely the interactions of such manifolds that are termed global bifurcations and cause in this case the loss of the quasiperiodic solution. 

Reactor type Bifurcation-distributive chip micro mixer ... 

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