Poincare surfaces of section

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]. 

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 17. The Poincare surface of section for T-shaped Hel2 with the initial vibrational state of I2 given by v = 10. Two bottlenecks to intramolecular energy transfer are shown, together with a 5 1 resonance zone and the dissociation dividing surface. From top to bottom the figures show how trajectories escape the first and then the second intramolecular botdenecks. The bottom panel shows trajectories passing the separatrix for dissociation. [From M. J. Davis and S. K. Gray, J. Chem. Phys. 84, 5389 (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 35. Poincare surface of section of 3-phospholene at = 5133 cm showing chaotic motion and embedded regions of quasi-periodic motion. [From C. C. Marston and N. De Leon, J. Chem. Phys. 91, 3392 (1989).]...

Figure 37. Poincare surface of section for HCN isomerization, showing both chaotic motion and quasi-periodic motion.

Furthermore, we investigate the detailed structure of the Poincare surface of section for the case of (Z, ) = (1,1). In this case, there are tori. These tori have the periodic points with period 6 in their outermost part. Here we counted the number of vertices of two triangle, namely 2x3 = 6. These periodic points is associated to one orbit in the whole phase space, which is an antisymmetric orbit in the configuration space. These periodic points have stable and unstable manifolds. In Fig. 8a, we depict the stable manifolds of the these periodic points. In Fig. 8b, we also depict the unstable manifolds by using the symmetry. The stable and unstable manifolds of these periodic points go to It should be noted that the reached points of them on 0 is the accumulation points of and Comparing Figs. 6a and 6b with Fig. 8, it is confirmed that in Figs. 6a and 6b is nearly parallel to the stable manifolds and in Fig. 6b is nearly parallel to the unstable manifolds. Therefore, it is understood that the foliated structure of tc and manifests the foliation of the stable and unstable manifolds—that is, hyperbolic structure. 

We are now ready to plot a Poincare surface of section for the double pendulum. We choose E = -2 and 19 different initial conditions defined... 

A Poincare surface of section may be used to identify the chaotic and quasi-periodic regions of phase space for a two-dimensional Hamiltonian. An ensemble of trajectories, chosen to randomly sample the phase space, are calculated and for each trajectory a point is plotted in the (9i,Pi)-plane every time Q2 = 0 for p2 > 0. A quasi-periodic trajectory lies on an invariant curve, while the points are scattered for a chaotic trajectory with no pattern. Figure 44 shows an example for a two-dimensional model for HOCl the HO bond distance is frozen in these calculations [351]. It clearly illustrates how the phase space becomes gradually more chaotic as the energy increases. 

Poincare surfaces of section are difficult to construct and to interpret for Hamiltonians with more than two degrees of freedom and other procedures must be used to identify a trajectory as quasi-periodic or chaotic. One approach is to calculate the power spectrum of a trajectory given by the Fourier analysis [352] according to... 

Figure 44 Poincare surfaces of section for a two-dimensional model of HOCl the HO bond distance is frozen in the classical calculations. 7 is the Jacobi angle and p is the corresponding momentum. Different symbols in square brackets denote different types of periodic orbits in the molecule. Reproduced, with permission of the American Chemical Society, from Ref. 258.

In two-dimensional Hamiltonian systems, the trajectories can be visualized by means of the Poincare surface of section plot. It is also possible to study two-dimensional Hamiltonian systems using the two-dimensional symplectic mapping. A typical phase space portrait of generic nonhyperbolic phase space is... 

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