Saddle connection

Notice that the twin saddle points are joined by a pair of trajectories. They are called heterocUnic trajectories or saddle connections. Like homoclinic orbits, heteroclinic trajectories are much more common in reversible or conservative systems than in other types of systems. ... 

Saddle connections) A certain system is known to have exactly two fixed points, both of which are saddles. Sketch phase portraits in which... 

In this broader context, what exactly do we mean by a bifurcation The usual definition involves the concept of topological equivalence (Section 6.3) if the phase portrait changes its topological structure as a parameter is varied, we say that a bifurcation has occurred. Examples include changes in the number or stability of fixed points, closed orbits, or saddle connections as a parameter is varied. 

Finally, when I < the saddle connection breaks and U spirals into the sink (Figure 8.5.9). 

Fig. 4.2 Phase portrait for logistic and bistable reaction terms. The front is a heterocUnic saddle-node connection for the logistic case. The front is a saddle-saddle connection for the bistable case...

Horizontal cylindrical vessels, supported by two or more saddles connected to a foundation platform. In this category many pressurized storage tanks and shell-and-... 

The other popular type of electrofusion joint is the service saddle joint. This is used to make tee connections between a small diameter service and a larger diameter main pipe. These were also originally heated tool type welds but electrofusion reduces operative involvement and provides a greater level of control in fusion conditions. The technology has also been extended to larger size saddle connections that produce pipe branches that can be made whilst the main pipe is still under pressure. 

Connections to the new liner will be needed once the liner installation is complete. This requires a portion of the old sewer be removed at the service connections. Once the liner is exposed, connection can be made. The connection to the liner is made with a heat-fnsion saddle or a strap-on saddle. If a strap-on saddle is used, it must be held in place with stainless steel clamps. A neoprene gasket is inserted between the liner and the strap-on saddle. Connections made to the saddle fittings are made with boots, fnll-encirclement clamps, or some other approved method. 

Fig. 8.1.6. (a) A structurally unstable saddle connection after the disappearance of a saddle-node cycle in Pig. 8.1.4 (b) Phase plane after the splitting of the homoclinic loop in Fig. 8.1.5. 

Van Strien, S. J. [1982] On Parameter Families of Vector Fields. Bifurcations Near Saddle-connections, Ph.D. Thesis, Utrecht University. 

The two homologous repeats, each of 88 amino acids, at both ends of the TBP DNA-binding domain form two stmcturally very similar motifs. The two motifs each comprise an antiparallel p sheet of five strands and two helices (Figure 9.4). These two motifs are joined together by a short loop to make a 10-stranded p sheet which forms a saddle-shaped molecule. The loops that connect p strands 2 and 3 of each motif can be visualized as the stirmps of this molecular saddle. The underside of the saddle forms a concave surface built up by the central eight strands of the p sheet (see Figure 9.4a). Side chains from this side of the P sheet, as well as residues from the stirrups, form the DNA-binding site. No a helices are involved in the interaction area, in contrast to the situation in most other eucaryotic transcription factors (see below). 

A transition state > 1 imaginary frequency The structure is a higher-order saddle point, but is not a transition structure that connects two minima. QST2 may again be of use. Otherwise, examine the normal modes corresponding to the imaginary frequencies. One of them will (hopefully) point toward the reactants and products. Modify the geometry based on the displacements in the other mode(s), and rerun the optimization. 

An IRC calculation examines the reaction path leading down from a transition structure on a potential energy surface. Such a calculation starts at the saddle point and follows the path in both directions from the transition state, optimizing the geometry of the molecular system at each point along the path. In this way, an IRC calculation definitively connects two minima on the potential energy surface by a path which passes through the transition state between them. 

An IRC calculation (indicates the two minima connected by the saddle point). 

At R > 400 pm the orientation of the reactants looses its importance and the energy level of the educts is calculated (ethene + nonclassical ethyl cation). For smaller values of R and a the potential energy increases rapidly. At R = 278 pm and a = 68° one finds a saddle point of the potential energy surface lying on the central barrier, which can be connected with the activated complex of the reaction (21). This connection can be derived from a vibration analysis which has already been discussed in part 2.3.3. With the assistance of the above, the movement of atoms during so-called imaginary vibrations can be calculated. It has been attempted in Fig. 14 to clarify the movement of the atoms during this vibration (the size of the components of the movement vector... 

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... 

Liquid-Fluid Equilibria Nearly all binary liquid-fluid phase diagrams can be conveniently placed in one of six classes (Prausnitz, Licntenthaler, and de Azevedo, Molecular Thermodynamics of Fluid Phase Blquilibria, 3d ed., Prentice-Hall, Upper Saddle River, N.J., 1998). Two-phase regions are represented by an area and three-phase regions by a line. In class I, the two components are completely miscible, and a single critical mixture curve connects their criticsu points. Other classes may include intersections between three phase lines and critical curves. For a ternary wstem, the slopes of the tie lines (distribution coefficients) and the size of the two-phase region can vary significantly with pressure as well as temperature due to the compressibility of the solvent. 

As explained above, the QM/MM-FE method requires the calculation of the MEP. The MEP for a potential energy surface is the steepest descent path that connects a first order saddle point (transition state) with two minima (reactant and product). Several methods have been recently adapted by our lab to calculate MEPs in enzymes. These methods include coordinate driving (CD) [13,19], nudged elastic band (NEB) [20-25], a second order parallel path optimizer method [25, 26], a procedure that combines these last two methods in order to improve computational efficiency [27],... 

Going in the opposite direction, i.e. when we consider the membrane stability with increasing ionic strength, we notice the approach of k towards zero. Going towards this value, the tendency of the bilayers to form saddle-shaped connections (also called stalks ) between bilayers increases. Saddle-shaped membrane structures also occur in processes like vesicle fusion, endo and exocytosis. The SCF predictions thus indicate that these events will occur with more ease at high ionic strength than at very low ionic strength. 

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