Hyperbolic point

In a recent analysis carried out for a bounded open system with a classically chaotic Hamiltonian, it has been argued that the weak form of the QCT is achieved by two parallel processes (B. Greenbaum et.al., ), explaining earlier numerical results (S. Habib et.al., 1998). First, the semiclassical approximation for quantum dynamics, which breaks down for classically chaotic systems due to overwhelming nonlocal interference, is recovered as the environmental interaction filters these effects. Second, the environmental noise restricts the foliation of the unstable manifold (the set of points which approach a hyperbolic point in reverse time) allowing the semiclassical wavefunction to track this modified classical geometry. 

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. 

Time-dependent, periodic, two-dimensional flows can result in streamlines that in one flow pattern cross the streamlines in another pattern, and this may lead to the stretching-and-folding mechanism that we discussed earlier, which results in very efficient mixing. In such flow situations, the outflow associated with a hyperbolic point can cross the region of inflow of the same or another hyperbolic point, leading, respectively, to homoclinic or heteroclinic intersections these are the fingerprints of chaos. 

Fig. 7.12 The flow pattern (bottom) is generated in a cavity filled with glycerin with the walls moving continuously in opposite directions. The flow pattern (top) has two elliptic and one hyperbolic point. [Reprinted by permission from J. M. Ottino, The Mixing of Fluids, Set Am., 20, 56-67, 1989.]...

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).]...

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. 

Figure 2.2 Streamlines of a meandering jet surrounded by recirculation zones showing typical structures in steady two-dimensional flows elliptic (E) and hyperbolic (H) stagnation points, flow regions with open and closed streamlines and separatrices that connect hyperbolic points. The flow is defined by the streamfunction ip(x, y) = Cy — tanh[(y — Acosx)/(Ly/1 + A2 sinx2)].

In the case of real eigenvalues, Ai 2 = A, x is called a hyperbolic fixed point, corresponding to a saddle point of the streamfunction, and it is of unstable character. It is located at the intersection of two special streamlines, called separatrices, which are the stable and unstable manifolds of the hyperbolic fixed point. These lines are defined as the set of points that approach the fixed point in the limits t —> Too, respectively. The motion around a hyperbolic point can be obtained as a solution of (2.40)... 

Thus in the neighborhood of hyperbolic points the distance between fluid particles, or the length of material lines, grows exponentially in time that would lead to efficient mixing. However, since the fluid elements are quickly ejected from the vicinity of these isolated points they only have a short term transient effect with little influence on the global mixing properties of the flow. 

Fluid elements are advected along closed circular streamlines, but the separatrices connecting the hyperbolic points inhibit long range advective transport from one cell to another (see Fig. 2.4). Thus... 

Figure 2.5 Sketch of the typical structures associated with the breakup of a resonant torus on the Poincare section in a weakly time-periodic flow producing a set of new elliptic and hyperbolic points.

Hydroperoxide Hypalon Hyperbolic flow Hyperbolic interfaces Hyperbolic point... 

This enables us to extract and visualize the stable and unstable invariant manifolds along the reaction coordinate in the phase space, to and from the hyperbolic point of the transition state of a many-body nonlinear system. PJ AJI", Pj, Qi, t) and PJ AJ , Pi, q, t) shown in Figure 2.13 can tell us how the system distributes in the two-dimensional (Pi(p,q), qi(p,q)) and PuQi) spaces while it retains its local, approximate invariant of action Jj (p, q) for a certain locality, AJ = 0.05 and z > 0.5, in the vicinity of... 

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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