Fixed points classification

As we ve seen, in one-dimensional phase spaces the flow is extremely confined— all trajectories are forced to move monotonically or remain constant. In higherdimensional phase spaces, trajectories have much more room to maneuver, and so a wider range of dynamical behavior becomes possible. Rather than attack all this complexity at once, we begin with the simplest class of higher-dimensional systems, namely linear systems in two dimensions. These systems are interesting in their own right, and, as we ll see later, they also play an important role in the classification of fixed points of nonlinear systems. We begin with some definitions and examples. 

By now you re probably tired of all the examples and ready for a simple classification scheme. Happily, there is one. We can show the type and stability of all the different fixed points on a single diagram (Figure 5.2.8). 

Points with a constant Mahalanobis distance from a fixed point are situated on a hyper ellipsoid which is defined by the covariance matrix (Figure 3). This distance measure is advantageously used for the classification of unknown objects. 

However, a similar classification of two-dimensional diffeomorphisms, or of three-dimensional fiows, is not that trivial. Let us illustrate this with an example. Consider a diffeomorphism T which has two saddle fixed points 0 and O2 with the characteristic roots )Ai) < 1 and i > 1 at (z = 1,2). Suppose that Wq and have a quadratic tangency along a heteroclinic orbit as shown in Fig. 8.3.1. The quadratic tangency condition implies that all similar diffeomorphisms form a surface of codimension-one in the space of all diffeomorphisms with a C -norm. 

Adequate data were available for development of the three AEGL classifications. Inadequate data were available for determination of the relationship between concentration and time for a fixed effect. Based on the observations that (1) blood concentrations in humans rapidly approach equilibrium with negligible metabolism and tissue uptake and (2) the end point of cardiac sensitization is a blood-concentration related threshold phenomenon, the same concentration was used across all AEGL time periods for the respective AEGL classifications. 

TTie classification of kinetic methods proposed by Pardue [18] is adopted in the software philosophy. TTie defined objective of measurement in the system is to obtain the best regression fit to a minimum of 10 data points, taken over either a fixed time (i.e. the maximum time for slow reactions) or variable time (for reactions complete in less than 34 min, which is the maximum practical observation time). In an analytical system generating information at the rate of SO datum points per second, with reactions being monitored for up to 2040 s, effective data-reduction is of prime importance. To reduce this large quantity of analytical data to more manageable proportions, an algorithm was devised to optimize the time-base of the measurements for each individual specimen. 

We may put the same point in terms of facts. Truth depends on facts. The worry is this. If facts are determined by the human mind, truth becomes subjective. But determination is ambiguous here between being a factor and being the only factor . The mind is indeed a factor. It fixes the criteria of individuation and classification, and thereby determines what sorts of facts there can be. As it were, it fixes the range of possible facts. But to do so is to do much less than fix the actual... 

By generalizing the idea of local convexity for any reference curvature value b [199], the number p(r,b) is the tool used for a classification of points r of the contour G(a) into various domains. For any fixed b, each point r of the contour surface G(a) belongs to one of three disjoint subsets of G(a), denoted by Aq, A, or A2, depending on whether at point r none, one, or both, respeetively, of the local canonical curvatures h and h2 are smaller than the reference value b [156]. The union of the three sets Aq, A, and A2 generates the entire contour surface, that is. 

A precise definition of the flowability of a powder is only possible with several numbers and curves, derived from a family of yield loci of the powder (measured with a shear cell) - see section 4 for further detail. Jenike23 proposed a simpler classification, according to the position of one point of the failure function (at a fixed value of the unconfined yield strength, say 5 lbf (22.3 N) with the Jenike shear cell, i.e. 3112 Pa or 65 lbf/ft2) with respect to the flow factor line (straight line through the origin, at a slope l///where//is the flow factor) - see Fig. 8 for a schematic representation of this. 

Although the classification of ternary phase behavior is described here at a fixed temperature, it is important to remember that a single ternary system can exhibit all three types of phase behaviors as the temperature of the system changes. Based on our classification of binary phase behavior, type-I ternary phase behavior above the critical temperature of the SCF solvent may revert to type-II or type-III ternary phase behavior if the operating temperature and pressure are adjusted to values near the critical point of the SCF solvent. 

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