Point singular

The total energy for a spherically symmetric radial configuration is 

Elastic anisotropy modifies the idealized configurations shown in fig. 3.5.10. More complex structures with an oblique orientation of the director at the surface have also been reported.  

Point singularities of equal and opposite strengths attract one another and are annihilated (see fig. 3.5.12). As the total energy of elastic deformation around a point defect increases linearly with the radius of the 

From the superposition principle (3.5.6) we know that the director pattern around a pair of like disclinations located at jc = d and — d is given by 

at all points (0,j ) on the midplane z = 0, the director orientation = sn+0Q = 6 . We may therefore conclude that a wall located at the midplane with the director firmly anchored on it at an angle 0 can be replaced by an image of the defect. Consequently the wall repels the defect with a force 

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Here the coefficients G2, G, and so on, are frinctions ofp and T, presumably expandable in Taylor series around p p and T- T. However, it is frequently overlooked that the derivation is accompanied by the connnent that since. . . the second-order transition point must be some singular point of tlie themiodynamic potential, there is every reason to suppose that such an expansion camiot be carried out up to temis of arbitrary order , but that tliere are grounds to suppose that its singularity is of higher order than that of the temis of the expansion used . The theory developed below was based on this assumption. 

In order to extend the existence of Eq. (137) for the singular points as well we write it as follows ... 

Figure 17. The differential closed paths F and the singular point B a, b) in the (p, q) plane (a) The point B is not surrounded by F. (b) The point B is surrounded by F.

Figure 19. The closed path F as a sum of three closed paths F, Fp, F,. (a) The closed (rectangular) paths, that is, the large path F and the differential path F both surrounding the singular point B(a, b). (6) The closed path Tp that does not surround the point fi(a, b). (c) The closed path F, that does not surround the point B[a,b).

Singular Points If/( ) is analytic in a region except at certain points, those points are called singular points. 

Phase Plane Singular Points.—We shall define the plane of the variables (x,y = x) as the phase plane and investigate the behavior of integral curves (or characteristics) in that plane by means of Eq. (6-2). In case we wish to associate with these curves the motion of the representative point R(x,y), we shall rather speak of them as trajectories and in this case one has to use Eq. (6-1). 

As dxfdt and dyjdt approach zero as R approaches the singular point, it is obvious that this approach is always asymptotic (i.e., occurs either for t- -co or for t - — oo. 

Singular points represent the positions of equilibrium of dynamical systems and merit further investigation. 

Elementary Singular Points.—One can start with a special... 

Consider next the case a < 0. Here the trajectories near the singular point, the saddle point, have the form shown in Fig. 6-3. Only four singular trajectories enter the saddle point (two of them, AS and BS, for t -> oo and two others, SD and SO, for t- - — oo). 

The third singular point, the focus, is obtained if one starts from the system... 

The trajectories are logarithmic spirals (Fig. 6-4). For a > 0, they wind on the singular point (i.e., the rotation of the radius vector is clockwise) for a < 0, they unwind (i.e., the rotation of the radius vector is counterclockwise). 

It can be shown that these singular points exist also for more general linear differential equations of the form... 

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. 

In other words, in normal cases the nature of equilibrium is determined only by the linear terms. This is also intuitively obvious since, as the trajectory approaches the singular point (at the origin), both x and y decrease indefinitely so that ultimately only the linear terms of the first order of magnitude remain. 

In applications the above properties of simple singular points are sufficient, and this yields very simple criteria. 

It must be noted that the singular point of the type center belongs to these special (or pathological ) cases. This case arises when the roots and S2 in the above terminology become purely imaginary the conditions axe then a + d = 0 6c > ad. ... 

We recall that in this terminology the center is the singular point (the state of rest) for simple harmonic motion represented in the phase plane by a circle (or by an ellipse). The trajectories in this case axe closed curves not having any tendency to approach the singular point (the center). 

Topological Configurations.—It may be noticed from the preceding that there is a certain relation between limit cycles and singular points. 

More specifically, from Fig. 6-5 it is observed that the trajectories reach the limit cycle from the inside on the other hand, in the theory of singular points we saw that when these points are unstable, trajectories leave them. 

If a half-trajectory C remains in a finite domain D without approaching singular points, the C is either a dosed trajectory or approaches such a trajectory. 

Suppose we have a certain topological configuration, say, SUS in our previous notation this means that the singular point is stable and the nearest cycle is unstable. The bifurcation of the first kind can be represented by the scheme ... 

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