Two Coupled Degrees of Freedom

Armed as we are now with the KAM theorem, the Center Manifold theorem, and the Stable Manifold theorem, we can begin to visualize the phase space of reaction dynamics. Returning to our original system (see Uncoupled Reaction Dynamics in Two Degrees of Freedom ), we now realize that the periodic orbit that sews together the half-tori to make up the separatrix is a hyperbolic periodic orbit, and it is not a fixed point of reflection. From our previous visualization of uncoupled phase-space dynamics, we know that the separatrix is completely nontwisted. In the terminology of Poliak and Pechukas, the hyperbolic periodic orbit is a repulsive PODS.  

There is now a division in phase space between trajectories that are instantaneously within the overlap between the two cylinders and those that are not. First of all, since the coupling is weak it is likely that most of the phase 

Poliak and Pechukas, and it can in a sense be thought of as a generalization of their work. These theories basically assume that motion within the chaotic region is statistical (although this assumption can sometimes be modified see below).29 

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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