Saddle map

The above result show that the concentration dependaiice of the Intensity maps is purely a statistical mechanics effect. In order to illustrate this important conclusion, we calculate disordered state, at concentration c=, with the V s obtained at the composition PtsV (figure 4). 150 K above the transition temperature, we Indeed observe the experimentally observed splitting of the diffuse intensity maxima, with a saddle point at (100). 

The lower a graph is more interesting. While initially the Poincar6 phase portrait looks the same as before (point E, inset 2c) an interval of hysteresis is observed. The saddle-node bifurcation of the pericxiic solutions occurs off the invariant circle, and a region of two distinct attractors ensues a stable, quasiperiodic one and a stable periodic one (Point F, inset 2d). The boundary of the basins of attraction of these two attractors is the one-dimensional (for the map) stable manifold of the saddle-type periodic solutions, SA and SB. One side of the unstable manifold will get attract to one attractor (SC to the invariant circle) while the other side will approach die other attractor (SD to die periodic solution). 

Figure 10. Divisions of the Em map for the quadratic Hamiltonian in Eq. (39). Va, Vb, and Vx are the energies of the two isomers and that of the saddle point between them. The solid lines are relative equilibria, and the cusp points, a , indicate points at which the secondary minimum and the saddle point in Fig. 9 merge at a point of inflection.

For a homonuclear diatomic molecule such as Cl2 the interatomic surface is clearly a plane passing through the midpoint between the two nuclei—in other words, the point of minimum density. The plane cuts the surface of the electron density relief map in a line that follows the two valleys leading up to the saddle at the midpoint of the ridge between the two peaks of density at the nuclei. This is a line of steepest ascent in the density on the two-dimensional contour map for the Cl2 molecule (Fig. 9). 

The definition of an atom and its surface are made both qualitatively and quantitatively apparent in terms of the patterns of trajectories traced out by the gradient vectors of the density, vectors that point in the direction of increasing p. Trajectory maps, complementary to the displays of the density, are given in Fig. 7.1c and d. Because p has a maximum at each nucleus in any plane that contains the nucleus (the nucleus acts as a global attractor), the three-dimensional space of the molecule is divided into atomic basins, each basin being defined by the set of trajectories that terminate at a given nucleus. An atom is defined as the union of a nucleus and its associated basin. The saddle-like minimum that occurs in the planar displays of the density between the maxima for a pair of neighboring nuclei is a consequence of a particular kind of critical point (CP), a point where all three derivatives of p vanish, that... 

Fig. 7.1 The electron density p(t) is displayed in the and

The top of the profile is maximum (saddle point) and is referred as the transition state in the conventional transition state theory. It is called a saddle point because it is maximum along the orthogonal direction (MEP) while it is minimum along diagonal direction of Fig. 9.12. The minimum energy path can be located by starting at the saddle point and mapping out the path of the deepest descent towards the reactants and products. This is called the reaction path or intrinsic reaction coordinate. 

The MM2 model resides very near the minimum 2 in the cellobiose energy map (cf. Fig. 9). (Among others, the crystal structure of methyl cellobioside-methanol complex is found in that minimum (15)). On the other hand, the PS79 model resides on the shallow saddle point between minima 2 and 3. 

The fractionation patterns exhibited % successive members of a progression of polyads (along 02, CC stretch, or along v4, trans-bend) provide a surveyor s map of IVR. One can look at the 1VR trends and see whether the multiresonance model expressed in the H nres (1 polyads provides a qualitative or quantitative representation of the fractionation patterns. The dynamics of even a four-atom molecule is so complicated that, unless one knows what to look for, one can neither identify nor explain trends in the dynamics versus V2 or u4 or Evib- Moreover, by defining the pattern of the IVR and how this pattern should scale with V2, v4, or EVib, the H res / polyad model may make it possible to detect a disruption of the pattern. Such disruptions could be due to a change in the resonance structure of the exact H near some chemically interesting topographic feature of the V(Q), such as an isomerization saddle point. 

The theory of bifurcations shows that the different types of bifurcations can be described in terms of normal forms, which represent local expansions of the dynamics around the bifurcating periodic orbit [19, 32, 49]. The purpose of the above mapping is to describe the successive bifurcations of the symmetric-stretch periodic orbit, starting from low energies above the saddle point. Appropriate truncation of the Taylor series of the potential v(q) around <7 = 0, which corresponds to the location of the symmetric-stretch orbit, provides us with the normal forms of the bifurcations [144], The bifurcations relevant for the dissociation dynamics under discussion can be described by truncating at the sixth order in q,... 

Tearpock DJ, and R. Bischke Applied Subsurface Geological Mapping, 2nd Edition. Prentice Hall. Inc, Upper Saddle River. NJ, 2002,... 

We now consider the phenomenon of entrainment (the development of resonances) on the torus (Meyer, 1983). When (and if) the off-diagonal band in Fig. 6 crosses the diagonal [Figs. 6(c) and 6(f)], there exist points whose images fall on themselves they are fixed points of the map we study. These points lie on periodic trajectories that are locked on the torus. Such trajectories appear in pairs in saddle-node bifurcations and are usually termed subharmonics . When this occurs there is no quasi-periodic attractor winding around the torus surface, but the basic structure of the torus persists the invariant circle is patched up from the unstable manifolds of the periodic saddle-points with the addition of the node-periodic point (Arnol d, 1973, 1982). As we continue changing some system parameter the periodic points may come to die in another saddle-node bifurcation (see Fig. 5). Periodic trajectories thus... 

FIGURE 5 Subharmonic saddle-node bifurcations, (a). The subharmonic period 3 isola for the surface model (o/o0 = 1.4, o0 = 0.001). One coordinate of the fixed points of the third iterate of the stroboscopic map is plotted vs. the varying frequency ratio oi/eio. Six such points (S, N) exist simultaneously, three of them (N) lying on the stable node period 3 and three (S) on the saddle period 3. Notice the two triple turning point bifurcations at o>/o>o = 2.9965 and 3.0286. In (b) the PFM of these trajectories on the isola Is also plotted, (c) shows another saddle-node bifurcation occurring (for a>o = 3) as this time the forcing amplitude is increased to o/o0 = I.6SS. 

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