Surface of section

A convenient method for visualizing continuous trajectories is to construct an equivalent discrete-time mapping by a periodic stroboscopic sampling of points along a trajectory. One way of accomplishing this is by the so-called Poincare map (or surface-of-section) method (see figure 4.1). In general, an N — l)-dimensional surface-of-section 5 C F is chosen, and we consider the sequence of successive in-... 

Microscope Study of the Surfaces of Sectioned M-17 Propellant Grains , PATR 2177 (1955) 23) S.M. Kaye, An Electron Microscope Study of the Surfaces of Sectioned M-15 Propeiiant Grains , PATR 2201 (1955) 24)S.M. Kaye,... 

Figure 2. (Left) Composite Poincare surface of section (SOS) for LiNC/LiCN at an excitation energy of E = 4600 cm-1. Two islands of regularity are seen corresponding to the two stable linear isomers, LiNC (0 = 180°) and LiCN (6 = 0). 

The classical dynamics of the FPC is governed by the Hamiltonian (1) for F = 0 and is regular as evident from the Poincare surface of section in Fig. 1(a) (D. Wintgen et.al., 1992 P. Schlagheck, 1992), where position and momentum of the outer electron are represented by a point each time when the inner electron collides with the nucleus. Due to the homogeneity of the Hamiltonian (1), the dynamics remain invariant under scaling transformations (P. Schlagheck et.al., 2003 J. Madronero, 2004)... 

Because the mapping approach treats electronic and nuclear dynamics on the same dynamical footing, its classical limit can be employed to study the phase-space properties of a nonadiabatic system. With this end in mind, we adopt a onemode two-state spin-boson system (Model IVa), which is mapped on a classical system with two degrees of freedom (DoF). Studying various Poincare surfaces of section, a detailed phase-space analysis of the problem is given, showing that the model exhibits mixed classical dynamics [123]. Furthermore, a number of periodic orbits (i.e., solutions of the classical equation of motion that return to their initial conditions) of the nonadiabatic system are identified and discussed [125]. It is shown that these vibronic periodic orbits can be used to analyze the nonadiabatic quantum dynamics [126]. Finally, a three-mode model of nonadiabatic photoisomerization (Model III) is employed to demonstrate the applicability of the concept of vibronic periodic orbits to multidimensional dynamics [127]. 

In order to identify the periodic orbits (POs) of the problem, we need to extract the periodic points (or fixed points) from the Poincare map. Adopting the energy F = 0.65 eV, Fig. 31 displays the periodic points associated with some representative POs of the mapped two-state system. The properties of the orbits are collected in Table VI. The orbits are labeled by a Roman numeral that indicates how often trajectory intersects the surfaces of section during a cycle of the periodic orbit. For example, the two orbits that intersect only a single time are labeled la and lb and are referred to as orbits of period 1. The corresponding periodic points are located on the p = 0 axis at x = 3.330 and x = —2.725, respectively. Generally speaking, most of the short POs are stable and located in... 

Figure 31. Periodic points of the Poincare map at the energy E = 0.65 eV. The Roman numerals indicate how often the corresponding orbit intersects the surface of section. Panel (b) shows an enlargement of the main regular island around x,p) = (3.3,0). The thin hnes represent various tori of the system.

Figure 8. Schematic representation of a chaotic repeller and its stable Ws and unstable IV manifolds in some Poincare surface of section (q,p) together with one-dimensional slices along the line L of typical escape time function T+1...

Such Hamiltonian mappings are generated by a Poincare surface of section transverse to the orbits of the flow. Thus, v(q) plays the role of a potential function for the motion perpendicular to the periodic orbit. Note that the mapping takes into account the nonseparability of the dynamics. 

In particular, the periodic orbits are in correspondence with finite sequences such as )W2 Up of period p. The periodicity occurring in the symbol sequences translates into the periodicity of the corresponding trajectory crossing the Poincare surface of section. 

Figure 15. Three-branch Smale horseshoe in the 2F collinear model of Hgl2 dissociation at the energy E = 600 cm 1 above the saddle in a planar Poincare surface of section transverse to the symmetric-stretch periodic orbit. The Smale horseshoe is here traced out in a density plot of the cumulated escape-time function (4.6).

Concerning the Poincare surface of section, it should be noticed that a sort of quantum surface of section can be constructed by intersection of the Wigner or Husimi transform of the eigenfunctions expressed in the quantum action-angle variables of the effective Hamiltonian, which can provide a comparison with the classical Poincare surfaces of section (e.g., in acetylene). 

The classical dynamics of a system can also be analyzed on the so-caUed Poincare surface of section (PSS). Hamiltonian flow in the entire phase space then reduces to a Poincare map on a surface of section. One important property of the Poincare map is that it is area-preserving for time-independent systems with two DOFs. In such systems Poincare showed that all dynamical information can be inferred from the properties of trajectories when they cross a PSS. For example, if a classical trajectory is restricted to a simple two-dimensional toms, then the associated Poincare map will generate closed KAM curves, an evident result considering the intersection between the toms and the surface of section. If a Poincare map generates highly erratic points on a surface of section, the trajectory under study should be chaotic. The Poincare map has been a powerful tool for understanding chemical reaction dynamics in few-dimensional systems. 

Figure 2. Poincare surface of section of the T-shaped Hel2 with a total energy -2662 cm . Q = R — Ro, where R is the He-l2 bond length and / o is its equilibrium value. P is the...

Figure 3. Poincare surface of section of the T-shaped HeBr2 at the Br-Br vibrational state V = 15, showing a 4 1 resonance. [From A. A. Granovsky et al., J. Chem. Phys. 108, 6282 (1998).]...

Figure 4. Poincare surface of section of molecular isomerization of cyclobutanone (C4H6O) for the total reaction energy E = O.Ola.u, showing a 3 1 resonance.

Figure 10. Poincare surface of section for collinear OCS relaxation at = 20,000 cm . It shows three major quasi-periodic regions, the resonance islands, the location of the dividing surface for intramolecular energy transfer, and a typical turnstile. [From M. J. Davis, J. Chem. Phys. 83, 1016 (1985).]...

Figure 14. An Hel2 surface of section for an unstable trajectory which forms a collision complex. The total energy is —2661.6 cm . Also shown are the reaction separatrix and the intramolecular bottleneck, (a) Graph showing the full dynamics of the trajectory. (b)-(f) Graphs illustrating the trajectory over five consecutive time ranges. These graphs are arranged to demonstrate the manner in which the trajectory moves with respect to the bottleneck and the separatrix. [From M. J. Davis and S. K. Gray, J. Chem. Phys. 84, 5389 (1986).]...

Figure 18. (a) A schematic composite surface of section for nomotating T-shaped Hel2. (b) Idealization of surface of section indicating flow out of various phase-space regions. The hatched areas represent regions of quasi-periodic motion. [From S. K. Gray, S. A. Rice, and M. J. Davis, J. Phys. Chem. 90, 3470 (1986).]... 

Figure 19. A schematic plot of the ideal bottlenecks on the Poincare surface of section for van der Waals molecule predissociation. R is the van der Waals bond length and P is the conjugate momentum. 5i is the intramolecular bottleneck dividing surface and S2 is the intermoleculear bottleneck dividing surface.

Figure 20. Schematic surface of section in modeling the Gray-Rice theory of isomerization, showing the separatrix and the phase regions A, B, and C, which are the generalized states of the system.

Figure 21. Construction of the exact separatrix on the surface of section for a symmetric double-well model potential, (a) The unstable manifold (b) Superposition of the stable (dashed) and unstable (solid) manifolds, (c) The exact separatrix, which is a union of portions of the above manifolds, (d) Turnstiles superimposed on the separatrix. [From S. K. Gray and S. A. Rice, J. Chem. Phys. 86, 2020 (1987).]...

Figure 28. Reactive island structure on a surface of section and kinetics data for the symmetric model Hamiltonian for E = 1.0. (a) The reactive island structure, (b) The population decay of isomer A from different calcualtions. [From A. M. O. De Almeida et al., Physica D 46, 265 (1990).]...

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