Periodic orbit dividing surfaces systems

Consider then an adiabatic well in the hyperspherical coordinate system. Classically, the motion of the periodic orbit at the well would be an oscillation from a point on the inner equipotential curve in the reactant channel to a point on the same equipotential curve in the product channel. This is qualitatively the motion of what are termed "resonant periodic orbits" (RPO s). For example the RPO s of the IHI system are given in Fig. 5. Thus, finding adiabatic wells in the radial coordinate system corresponds to finding RPO s and quantizing their action. Note that in Fig. 5 we have also plotted all the periodic orbit dividing surfaces (PODS) of the system, except for the symmetric stretch. By definition, a PODS is a periodic orbit that starts and ends on different equi-potentials. Thus the symmetric stretch PODS would be an adiabatic well for an adiabatic surface in reaction path coordinates. However, the PODS in the entrance and exit channels shown in Fig. 5 may be considered as adiabatic barrieres in either the radial or reaction path coordinate systems. Here, the barrier in radial coordinates, has quantally a tunneling path between the entrance and exit channels. 

We also found [41] that besides total energy, the velocity across the transition state plays a major role in many-dof systems to migrate the reaction bottleneck outward from the naive dividing surface S(qi = 0). A similar picture has been observed by Pechukas and co-workers [37] in 2D Hamiltonian systems, that is, as energy increases, pairs of the periodic orbit dividing surfaces (PODSs) appearing on each reactant and product side migrate outwards, toward reactant and product state, and the outermost... 

Similar to unstable periodic orbits, an NHIM has stable and unstable manifolds that are of dimension 2 — 2 and are also structurally stable. Note that a union of the segments of the stable and unstable manifolds is also of dimension 2n — 2, which is only of dimension one less than the energy surface. Hence, as far as dimensionality is concerned, it is possible for a combination of the stable and unstable manifolds of an NHIM to divide the many-dimensional energy surface so that reaction flux can be dehned. However, unlike the fewdimensional case in which a union of the stable and unstable manifolds necessarily encloses a phase space region, a combination of the stable and unstable manifolds of an NHIM may not do so in a many-dimensional system. This phenomenon is called homoclinic tangency, and it is extensively discussed in a recent review article by Toda [17]. 

One can easily verify that Eqs. (A.167) and (A.168) satisfy Eq. (A.6)). To pass through the dividing surface from the one side to the other, the crossing trajectories typically require, at most, only a half period of the reactive hyperbolic orbit nl o >p. Furthermore, in the regions of first-rank saddles, such passage time intervals are expected to be much shorter than a typical time of the systems to find the variation or modulation of the frequency oop with respect to the curvature, that is, n cbp = (9 s )) = d(s )). 

The procedure for studying a fc-parameter family is similar to that for the one-parameter case firstly, divide the space of the parameters p into the regions of topologically equivalent behavior of trajectories, and study the system in each of these regions. Secondly, describe the boundaries of these regions (the bifurcation set), and finally study what happens at the bifurcation parameter values. We will see below that in the simplest cases (e.g. an equilibrium state with one zero or a pair of imaginary characteristic exponents, or a periodic orbit with one multiplier equal to 1 or to —1) one can almost always, except for extreme degeneracies, choose a correct bifurcation surface of a suitable codimension and analyze completely the transverse families. Moreover, all of these families turn out to be versal. 

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