Bifurcation mechanisms

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.

Scheme 22.4 A bifurcated mechanism accounting for the effect of basic additives.

A wide class of analytic second-order phase transitions is characterized by their Landau bifurcational mechanism [38]. According to this mechanism, a system characterized by order parameter r], possesses a single stable equilibrium solution (rje = 0) for a range of the external parameter T (T > Tcr see a schematic illustration in Fig. 2.3.4a). This single solution corresponds to an absolute internal minimum of the system s free energy F as a function of the order parameter (Fig. 2.3.4b, Curve 1). As the external parameter T decreases, at a critical value T = Tcr, the solution with r)e = 0 becomes unstable with two more stable solutions with r e 0 (for T < TCI) bifurcating from it (Fig. 2.3.4a). In the (F, rf) plane this corresponds to the appearance of two new local free energy minima that flank the former one, which now turns into a local maximum (Fig. 2.3.4b, Curve 2). 

Singamaneni S, Bertoldi K, Chang S, Jang JH, Young SL, Thomas EL, Boyce MC, Tsukruk VV (2009) Bifurcated mechanical behavior of deformed periodic porous solids. Adv Funct Mater 19 1426-1436... 

The important case of specimens with a fixed total thickness was considered in Ref. [40]. There are certain features of crack bifurcation under these conditions, such as that if the sample with a fixed total thickness has too large a number of layers there will be no bifurcation. Layer thickness and composition are important and efficient parameters to control the bifurcation in laminates. The effect is comparable with a crack bridging phenomenon [21], The bifurcation mechanism increases the laminate fracture toughness by approximately 1.5-2 times. 

This was the first complete structure of the bc complex. The structure provided information about all 11 subunits and revealed that subunit 9, the mitochondrial targeting presequence of ISP, exists between two core subunits, which are most likely a mitochondrial targeting presequence peptidase. We have solved the structures of the bc complex in two different crystal forms. Surprisingly, the conformation of the Rieske FeS protein was totally different between two crystal forms, and this provided a crucial insight of the electron bifurcation mechanism at the Qp site (see Section II,F). 

Figure 10 shows the proposed ubiquinol oxidation and electron bifurcation mechanism at Qp site. (A) In the absence of the ubiquinone, the side chain of Glu-271 is connected to the solvent in the mitochondrial intermembrane space via a water chain. (B) As a reduced ubiquinol molecule binds to the site, the side chain of Glu-271 flips to form a hydrogen bond to the bound ubiquinone. (C) Now, the ISP, which is moving around the intermediate position by thermal motion is trapped at the b" position by a hydrogen bond to the bound ubiquinone. (D,E) Coupled to deprotonation, the first electron transfer occurs. Since the Rieske FeS cluster has a much higher redox potential (ca. +300 mV) than heme bl (ca. 0 mV), the first electron is favorably transferred to ISP. This yields ubisemiquinone, (F,G). After ubisemiquinone formation, the hydrogen bond to the His-161 of ISP is destabilized. The ISP moves to the c position, where the electron is transferred from the Rieske FeS cluster to heme c. Now unstable ubisemiquinone is left in the Qp pocket. The redox potential of the deprotonated ubisemiquinone is assumed to be several hundred millivolts. Now the electron transfer to the heme bl is a downhill reaction. (H) Coupled to the second electron transfer, the second proton is transferred to Glu-271 and subsequently to the mitochondrial intermembrane space. The fully oxidized ubiquinone is released to the membrane. 

Fig. 10. Possible electron bifurcation mechanism at the Qp site. See text for details.

The bifurcation mechanisms for formation of multi-slip fault zones suggest that maximum fault zone thickness will often correspond to the strike-normal distance between the traces of two overlapping slip surfaces (Fig. 2c). Fault overlaps and their breached equivalents occur on faults of all sizes as do, by implication, paired and multi-slip surface fault zones. Complex and paired slip surface fault zone structures will occur on scales below that resolvable by even high quality seismic data (lateral resolution is no better than 50-100 m at North Sea reservoir depths). The possible impact of sub-seismic complexity and paired slip surfaces on connectivity and sealing across faults offsetting an Upper Brent type sequence are briefly considered below. 

Let us clarify the underlying bifurcation mechanisms, which cause the observed phenomenon. The inspection of the system dynamics and the continuation technique reveal that the blowup of the synchi onous attractor at j = 0.0284 occurs via a collision with a chaotic saddle. The chaotic saddle is located in the phase space as shown in Fig. 6.19(b). In order to detect the geometrical place of this unstable chaotic saddle, we used the continuation technique. We follow ed the unstable low-periodic orbits, which were born in the period-doubling cascade. 

Scheme 2. Kochi s catalytic cycle (lel t) for the Mn "(salcn)/PhlO catalytic system and Adam-Collman s bifurcated mechanism (right) [30,311...

Let X be a set of parameter values for which a solution of eqs. (2), referred to as reference state, loses its stability and gives rise to new branches o7 sol uti ons by a bifurcation mechanism. We want to see how the solution of the master equation, eq. (1), behaves under these conditions, and how this behavior depends on small changes of the parameters X around The answer to this question depends on the kind of bifurcation considered, on the nature of the reference state, and on the number of variables involved in the dynamics. The simplest case is, by far, the pitchfork bifurcation occurring as a first transition from a previously stable spatially uniform stationary state. This transition is characterized by a remarkable universality. First, whatever the number of variables present initially, it is always possible to cast the stochastic dynamics in terms of a single, "critical" variable. This is the probabilistic analog of adiabatic elimination or, in more modern terms, of the center manifold theorem [4,8-10]. Second, the stationary probability distribution of the critical variable can be cast in the form (we set 6X = (p-Xg, rstands for the spatial coordinates) ... 

A similar theoretical analysis can be reproduced for higher dimensional systems [48]. In a prospective paper [63], in collaboration with Pearson and Russo, we have reported preliminary results of a study of two-dimensional reaction-diffusion systems that model sustained front patterns observed in gel reactors. The linear [34, 35] and annular [21, 22, 38] gel reactors are strips of gel that are fed from the lateral boundaries. These reactors have a natural tendency to produce narrow front (linear or circular) structures away from the boundaries [39]. Stationary single-front and multi-front patterns have been observed experimentally [34-38]. According to the Hopf bifurcation mechanism reported in section 5, these front patterns are expected to destabilize into periodically oscillating structures. Since the Hopf mode is likely to be condensed in the active region at the front zones, the reaction-diffusion sys-... 

Figure 9 Maps of the gradient vector field of the electron density along the symmetrical dissociation path for water illustrating the bifurcation mechanism of structural change. Also shown for structures a. b. and c are profiles of the electron density along the C2 symmetry axis. In a there are no cps along this axis in b there is a single degenerate cp where both first and second derivatives of p(r ) varysh in c there are two stable cps, the maximum associated with the H-H bond cp, the minimum with the ring cp...

These examples have identified two types of catastrophe points, a distinction that arises as a corollary of a theorem on structural stability. This theorem, when used to describe structural changes in a molecular system, states that the structure associated with a particular geometry X in nuclear configuration space is structurally stable if p r X) has a finite number of cps such that (i) each cp is nondegenerate (ii) the stable and unstable manifolds of any pair of cps intersect transver-sally. The immediate consequence of this theorem is that a structural instability can be established solely through either of two mechanisms in the bifurcation mechanism the charge distribution exhibits a degenerate cp, while the conflict mechanism is characterized by the nontransversal intersection of the stable and unstable manifolds of cps in p(r X). 

Table 1 Character Table for C(48), the Appropriate Molecular Symmetry Group for the Water Trimer if Both the Single-Hip and Bifurcation Mechanisms are Feasible. The Number of Elements in Each Class is Given in Each Case and Representative Operations are Shown in Table 2. = exp( >/3)...

Figure 5 (a) Single-flip and (b) bifurcation mechanisms calculated... 

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