Stability Analysis of Periodic Orbits

We have discussed the typical manifestation of periodic orbits on a Poincare map as fixed points that are either elliptic or hyperbolic. Let us now consider the properties of motion nearby these fixed points in terms of their stability properties. This is accomplished by a straightforward linear stability analysis about the fixed point. We can carry out such an analysis on any fixed point, whether or not the surrounding phase space is chaotic (as long as we can find the fixed point). 

In considering dynamics on the Poincare map, it is extremely useful to think of the dynamics as being generated by a mapping U  

Another, slightly more technical ( ) way to understand area preservation is to recall that the coordinates of all the points on the Poincare map are specified by coordinates that are canonically conjugate. Because both the initial and final coordinates of a family of trajectories propagated for one mapping are so specified, there must exist a generating function that transforms the coordinates of the initial points into those of the final points. Such a generating function is necessarily a canonical transformation. All canonical transformations preserve the norm of the vectors they transform it can be shown that this property is equivalent to area preservation on the Poincare map.  

If we assume that the final coordinates are a linear function of the displacement of ip2, 2) froni the fixed point (p% we have 

In matrix form this can easily be shown to be the same as  

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ABSTRACT. The study of periodic orbits embedded in the continuum has provided a new tool for understanding the dynamics of molecular collisions, The application of periodic orbit theory to classical variational transition state theory, quantal threshold and resonance effects is presented. Special emphasis is given to the stability analysis of periodic orbits in collinear and three dimensional systems. Future applications of periodic orbit theory are outlined. 

One of the theoretical shortcomings of the periodic reduction method is the rather arbitrary definition of the bend angle. Although, as was the case to date, one may a posteriori verify that the adiabatic reduction is justified by comparing bend frequencies with stretch frequencies, it is still desirable to construct a method which a priori does not have this ambiguity. One possibility is using the quasiperiodic reduction method outlined in this section. However, as we shall point out a much simpler method may be based on stability analysis of periodic orbits in 3D. 

It seems to me that stability analysis of periodic orbits is a powerful tool which has yet to be fully utilised. The discussion in sec. IV.c is really more a research proposal than a review. No one has yet studied the stability of periodic orbits in 3D systems. The study presented in this paper is simplified since only collinear like potential energy surfaces were used. What happens in a system like LiFH whose minimum energy path is certainly not collinear ... 

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