Focus nodes

The three standard local codimensional-one bifurcations are the saddle-node, Hopf, and period doubling bifurcations and several have been continued numerically for this model and appear in figure 2. We have chosen not to show the curves of focus-node transitions because they do not represent any changes in stability, only changes in the approach to the steady behaviour. The saddle-node bifurcations that occur during phase locking of the torus at low amplitudes continue upward and either close upon themselves as in the case of the period 3 resonance horns or the terminate in some codimension-two bifurcation. 

Let us now examine the behaviour of the solutions for the dynamic system (20) in time and analyze the system trajectories in the phase pattern. This analysis permits us to characterize peculiarities of the unsteady-state behaviour (in particular to establish whether the steady state is stable or unstable), to determine its type (focus, node, saddle, etc.) and to find attraction regions for stable steady states, singular lines, etc. 

Figure 26. Skeleton bifurcation diagram in the t/-p parameter plane for the model equation (16). Shown are Hopf and saddle-node bifurcations (SUN = saddle-unstable-node bifurcation) as well as the border of the focus-node transition (dashed line) mixed-mode wave forms exist close to the dark region (which marks the region where a fixed point is a ShQ nikov saddle focus). The phase portraits sketch the Unear stability of the fixed point(s). (Reprinted with permission from M. T. M. Koper and P. Gaspard, J. Chem. Phys. 96, 7797, 1992. Copyright 1992, American Institute of Physics.)...

As mentioned before, a shape is a group of constraints that can be validated against RDF nodes. In order to decide which nodes (henceforth called focus nodes) shall be considered for validation, shapes can have different types of scopes that instruct a SHACL processor on how to select them, namely ... 

Input graph Selected focus nodes Remaining focus nodes Constraints produce... 

Inverse) Property Constraints A (inverse) property constraint (sh property, sh inverseProperty) is a constraint that defines restrictions on the values of a given property in the context of the focus node. The focus node is the subject (or object for inverse property constraints) and the property represents the predicate of respective triples (sh predicate). Each (inverse) property constraint must constrain exactly one property at a time. An example of such a property constraint is defined in sh ComputerShape illustrated in Listing 13.2. It states that every individual of type ol Computer must have exactly one (sh minCount,sh maxCount) property value for property oLhasID and that this one property value must be of type xsd string (sh datatype). ... 

Node Constraints A node constraint (sh constraint) can be used to define constraints about the focus node itself rather than on values of its properties. 

Cardinality CC e.g., for restricting the minimum number of allowed property values for certain properties of a focus node (sh minCount). 

Property Pair CC e.g., for validating whether each of the value sets of a given pair of properties at the same focus node are equal (shiequal). 

Note, that most CC can be applied within multiple contexts, hence the concept of value nodes either refers to objects, subjects or the focus node itself depending on the context its respective CC is used in. 

HAZOP focuses on study nodes, process sections, and operating steps. The number of nodes depending on the team leader and study objectives. Conservative studies consider e er line and vessel. An experienced HAZOP leader may combine nodes. For example, the cooling looser . .ater chlorination system may be divided into a) chlorine supply to venturi, b) recirculation loop, and e) to .er water basin. Alternatively, two study nodes may be used a) recirculation loop and tower water basin, and b) chlorine supply to venturi. Or one study node for the entire process. 

Since singular points are identified with the positions of equilibria, the significance of the three principal singular points is very simple, namely the node characterizes an aperiodically damped motion, the focus, an oscillatory damped motion, and the saddle point, an essentially unstable motion occurring, for instance, in the neighborhood of the upper (unstable) equilibrium position of the pendulum. 

The ocean plays a central role in the hydro-spheric cycling of sulfur since the major reservoirs of sulfur on the Earth s surface are related to various oceanic depositional processes. In this section we consider the reservoirs and the fluxes focusing on the cycling of sulfur through this oceanic node. 

If patients are hemodynamicaUy stable, the focus should be directed toward control of ventricular rate. Drugs that slow conduction and increase refractoriness in the AV node should be used as initial therapy. In patients with normal LV function (left ventricular ejection fraction >40%), IV j3-blockers (propranolol, metoprolol, esmolol), diltiazem, or verapamil is recommended. If a high adrenergic state is the precipitating factor, IV /J-blockers can be highly effective and should be considered first. In patients with left ventricular ejection fraction <40%, IV diltiazem and verapamil... 

These reviews can be either in addition to or combined with periodic process hazard analyses (PHAs) by using methods such as what-if analysis and HAZOP studies. The latter should consciously focus on identifying scenarios in which intended reactions could get out of control and unintended reactions could be initiated. One means of accomplishing this as part of a HAZOP study has been to include chemical reaction as one of the parameters to be investigated for each study node. Johnson and Unwin (2003) describe other PHA-related approaches for studying chemical reactivity hazards. 

Figure 29 Bifurcation diagram of the minimal model of glycolysis as a function of feedback strength and saturation 6 of the ATPase reaction. Shown are the transitions to instability via a saddle node (SN) and a Hopf (HO) bifurcation (solid lines). In the regions (i) and (iv), the largest real part with in the spectrum of eigenvalues is positive > 0. Within region (ii), the metabolic state is a stable node, within region (iii) a stable focus, corresponding to damped transient oscillations.

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