Poincare surface, Hamiltonian mapping

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. 

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. 

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... 

Abstract The periodic orbits play an important role in the study of the stability of a dynamical system. The methods of study of the stability of a periodic orbit are presented both in the general case and for Hamiltonian systems. The Poincare map on a surface of section is presented as a powerful tool in the study of a dynamical system, especially for two or three degrees of freedom. Special attention is given to nearly integrable dynamical systems, because our solar system and the extra solar planetary systems are considered as perturbed Keplerian systems. The continuation of the families of periodic orbits from the unperturbed, integrable, system to the perturbed, nearly integrable system, is studied. 

Figure 12 (Top) two trapped quasiperiodic orbits at fixed energy superimposed on potential energy contours for the De Leon-Berne Hamiltonian at = 0.65 (see Figure 23 for the potential energy surface). (Below) The Poincare map for this system at this energy, with the surface of section defined at fixed <]2 with p2 > 0. The trajectories above are connected with their corresponding map locations below. Note that whereas these relatively regular motions lie on tori, most of the Poincare map is broken up, indicating that most motions at this energy are chaotic. Reprinted with permission from Ref. 119.

Although the De Leon-Berne Hamiltonian is apparently the only system for which these fairly complex surface plots have been made, the reactive islands Poincare map structure (which is a unique signature of the cylindrical geometry) has been observed in models of 3-phospholene as well as in a symmetric triple-well prototype. 24 ij has also been observed in the bi-molecular reactions H + H2 - H2 + H52,i24 anj fn (he unimolecular decomposition of the (He I2) cluster - and of HNSi. 24 These studies have shown that even in a strong-coupling limit, where the repulsive PODS wanders away from the barrier to some extent or even bifurcates into multiple P0DS, > - 124 cylindrical separatrix manifolds mediate the pre- and postreaction dynamics. 

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