Elliptic point

For case (a), the flow field consists of a single convective cell which circulates around an elliptic point (center) located at X= Y= 0 [28], For case (b) the flow field consists of two counter-rotating cells with centers at X = 0 and Y = 0.58. The cells are separated by the surface Y= 0. For case (c), the flow field is similar to (b) in the sense that the flow field consists of two counter-rotating cells separated by the surface at X= 0. The centers of rotation are at X= 1 and Y = 0. For case (d), the flow field consists of four counter-rotating cells separated by two surfaces at X = 0 and Y = 0. 

For larger T (T = 1.6), chaotic behavior arises, the hyperbolic fixed point is disrupted and the tori are perturbed (see Figure 1.20) [28]. A chaotic region appears with homoclinic tangle and formation of new hyperbolic and elliptic points. 

In the no-barrier zone, the cross-sectional flow field helical flow is induced by the grooves and shows non-linear rotation with only one elliptic point [58], In the barrier zone, a spatially periodic perturbation on the helical flow is imposed and thereby two co-rotating flows form, characterized by a hyperbolic point and two elliptic points. By periodic change of the two flow fields, a chaotic flow can be generated. 

At the same time Chien, Rising, and Ottino (17) studied chaotic flow in two-dimensional cavity flows with a periodic moving wall, which is relevant to mixing of viscous polymeric melts. All two-dimensional flows, as pointed out by Ottino (18), consist of the same building blocks hyperbolic points and elliptic points. A fluid particle moves toward a hyperbolic point in one direction and away from it in another direction, whereas the fluid circulates around parabolic points, as shown in Fig. 7.12. 

Fig. 7.14 Elliptic and hyperbolic points in Fig. 7.13(d). Circles represent elliptic points, and squares, hyperbolic points. [Reprinted by permission from J. M. Ottino, C. W. Leong, H. Rising, and R D. Swanson, Morphological Structures Produced by Mixing in Chaotic Flows, Nature, 333, 419 125 (1988).]...

End view cross-sectional streamlines Hyperbolic and elliptic points move... 

In this section we provide an analjdical proof for the absence of regular period-1 islands in the phase space of the positively kicked onedimensional hydrogen atom. This proof may also serve as a template to prove the absence of elliptic points for other important atomic physics systems such as the stretched hehum atom discussed in Chapter 10. 

In the vicinity of elliptic points, the surface can be fitted to an ellipsoid, whose radii of curvature are equal to those at that point (Fig. 1.5a). The surface lies entirely to one side of its tangent plane, it is "synclastic" and both curvatures have the same sign. About a parabolic point, the surface resembles a cylinder, of radius equal to the inverse of the single nonzero principal curvature (Fig. 1.5b). H3rperbolic ("anticlastic") points can be fitted to a saddle, whic is concave in some directions, flat in others, and convex in others (Fig. 1.5c). At hyperbolic points, the surface lies both above and below its tangent plane. 

For this value of the perturbing parameter a lot of orbits are still invariant tori. Some resonant curves are displayed surrounding some elliptic points and a chaotic, though well confined, zone is generated by the existence of the hyperbolic point at the origin. 

Let us consider the elliptic point (1.239,0) of the standard map (equation 5) for e = 0.7, which corresponds to a periodic orbit of order 4. We have computed the evolution of the FLI with time for ten orbits regularly spaced on the x-axis in the interval [1.239,1.339] starting from the periodic point. 

Thus there are two qualitatively different cases. When the eigenvalues are complex, Ag2 = iuj, x is an elliptic fixed point corresponding to a local maximum or minimum of the streamfunction. Such elliptic points are surrounded by closed circular orbits that stay in the close vicinity of x. Therefore the area around the elliptic points remains isolated from the rest of the flow, thus preventing efficient mixing. 

According to the Poincare-Birkhoff fixed point theorem all resonant tori break up for arbitrarily small perturbations. If the rotation number is p/q the perturbation leaves q pairs of hyperbolic and elliptic periodic orbits. The unstable hyperbolic orbits are embedded in a layer filled by aperiodic, chaotic orbits that do not stay on an invariant surface, but cover a finite non-zero volume of the phase space in a chaotic layer around the original resonant torus. The elliptic points, however, are wrapped around by new concentric tori that form islands of regular orbits within the chaotic band (Fig. 2.5). 

The experiments also demonstrated presence of the mixing islands , where very little mixing took place. In consequence, the chaotic mixing could be schematically represented by the streamline diagram comprising the elliptic points located in the center of the blinking vortex and a hyperbolic... 

When 6=1, the Henon map becomes conservative, as its Jacobian equals -1-1. At 6 = 1 and a = —1, it has an unstable parabolic fixed point with two multipliers +1 at 6 = 1 and a = 3, it is a stable parabolic fixed point with two multipliers —1. In between these points, for —1 < a < 3 (i.e. (a, 6) G T), the map has a fixed point with multipliers where cos > = 1 y/a -h 1. This is a generic elliptic point for tp 7r/2,27r/3,arccos(—1/4) [167]. Since the Henon map is conservative when 6=1, the Lyapunov values are all zero. When we cross the curve AH, the Jacobian becomes different from 1, hence the map either attracts or expands areas which, obviously, prohibits the existence of invariant closed curves. Thus, no invariant curve is born upon crossing the curve AH. ... 

FIGURE 6.26 Elliptic and hyperbolic points. A blinking vortex system with vortex centers at the elliptic points can produce this streamline pattern. 

The definition of the homoclinic and heteroclinic points needs first the introduction of hyperbolic and elliptic points. A two-dimensional flow always consists of hyperbolic and/or elliptic points (Fig. 6.26). At the hyperbolic point the fluid moves toward it in one direction and away from it in another direction. At an elliptic point the fluid moves in closed pathlines. A periodic point is defined as the point at which a particle in a periodic flow returns after a number of periods. The number of periods defines also the order of the periodic point, as periodic point of period 1, 2, and so on. Note that the periodic elliptic points should be avoided should we want enhanced mixing. A point where the outflow of one hyperbolic point intersects the inflow of another hyperbolic flow is called transverse heteroclinic point. When the inflow and outflow refer to the same hyperbolic point, the point is called transverse homoclinic point. 

The second order tensor in 2D images was also used by Noble [31], who pointed out that the local image surface can be classified according to the Hessian matrix determinant as a planar point (zero determinant), parabolic point (zero determinant), hyperbolic point (negative determinant) and elliptic point (positive determinant). Points of interest are those which contain strong intensity variation such as the hyperbolic and elliptic points. 

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