Quasi-periodic orbits

D. Quasi-periodic Orbit Analysis of Photoisomerization Dynamics... 

Although the phase space of the nonadiabatic photoisomerization system is largely irregular, Fig. 36A demonstrates that the time evolution of a long trajectory can be characterized by a sequence of a few types of quasi-periodic orbits. The term quasi-periodic refers here to orbits that are close to an unstable periodic orbit and are, over a certain timescale, exactly periodic in the slow torsional mode and approximately periodic in the high-frequency vibrational and electronic degrees of freedom. In Fig. 36B, these orbits are schematically drawn as lines in the adiabatic potential-energy curves Wo and Wi. The first class of quasi-periodic orbits we wish to consider are orbits that predominantly... 

Figure 36, (A) Classical time evolution of the reaction coordinate (p as obtained for a representative trajectory describing nonadiabatic photoisomerization. (B) The vibronic motion of the system can be characterized by various quasi-periodic orbits, which are schematically drawn here as lines in the adiabatic potentials Wo and Wi. (C) As an example of a mixed orbit, the time evolution between t — 2.7 ps and t2 =3.9 ps is shown here in more detail.

To summarize, we have characterized the nonadiabatic photoisomerization dynamics in terms of a few quasi-periodic orbits. These orbits are close to an... 

In the liner approximation, we see thus that the NHIM is made of periodic/ quasi-periodic orbits, organized in the usual tori characteristic of the integrable systems. Because the NHIM is normally hyperbolic, each point of the sphere has stable/unstable manifolds attached to it. This situation is exactly parallel to the one described earlier for PODS. The equation for it is... 

One of the phenomena that is unique in systems with many degrees of freedom is that the state of the system can undergo large changes regardless of the existence of invariant sets that are remnants of periodic/quasi-periodic orbits of the unperturbed system [1], Arnold showed this phenomenon for a specific model (shown in Section IE) and calculated the upper bound of the rate of change in the action variable by linear perturbation theory. This phenomenon is sometimes called Arnold diffusion [2]. 

We have also used the periodic reduction method to predict with good accuracy the 3D structure of vibrationally bonded molecules. It should be stressed though, that in principle it is not necessary to use periodic reduction. As shown in Fig. 9 the RPO s of the IHI system are stable also in 3D, one can find bound quasi-periodic orbits and quantize them semiclassically directly without resorting to periodic reduction. 

The elimination of secular terms from the power series expansion of the solution is achieved by the method of Lindstedt. The underlying idea is to pick a fixed frequency p, and to look for a quasi-periodic solution with basic frequencies /i and v. This is actually the same thing as looking for a quasi-periodic orbit on an invariant 2-dimensional torus. The process of solution is the following. Write the Duffing s equation as... 

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