Stable/unstable manifolds

While the one-dimensional case may seem too simple, even trivial, it presents a good opportunity to put forward some very general concepts. These concepts, like the existence of barriers in phase space and the stable/unstable manifolds theorem, are best introduced here, having in mind that most interesting applications will come later on. Also, the one-dimensional case has been employed in less trivial ways, by reducing all rapid DOFs to some adiabatic approximation allowing nonlinear one-dimensional TST to be applied [34]. 

The local and global stable unstable manifold theorems (see, e.g.. Ref. 24, pp. 136-140) tell us the following are the (un)stable manifolds) ... 

Figure 6. The nonlinear stable/unstable manifolds are tangent to the linearized stable/unstable manifolds.

Cylindrical Manifolds. There is one big advantage of looking at 2-DOF TS in phase space It puts emphasis on the existence of the tubes that determine the transport of classical probability in phase space. Existence of those tubes has been known for a long time [48]. These tubes are the set of trajectories that constitute the stable/unstable manifolds of PODS. Locally, in the vicinity of P, they immediatly generalize to higher dimensions. They are constructed as follows ... 

Integrate the trajectory along this direction one finds the stable/unstable manifold itself. 

And finally, the TS itself is found at the intersection of tubes. More precisely, the TS (which is two-dimensional in the three-dimensional energy level) is found at the intersection of the interiors of the two tubes, constituted by the stable/unstable manifolds, at each side of the T PODs. A rigorous definition (valid for n DOF) is found in Ref. 9. An illustration may help the intuition (Fig. 11). 

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... 

Transport. We need now to construct the NHIM, its stable/unstable manifolds, and the center manifold. Let P be the main relative equilibrium point. The first task is to find the short periodic orbits lying above P in energy. These p.o. are unstable. We did so by exploring phase space at energies 4, 10, and 14 cm above E (1 atomic unit = 2.194746 x 10 cm ). It is not possible to go much higher in E, since the center manifold disappears shortly above E + 14cm , because of the structure of the potential energy surface. 

Figure 22. Two views of the stable/unstable manifolds of P, for 7 = 8. The inner part and outer part of phase space are clearly seen. They cross at the TS.

The set of all points of the phase space whose trajectories converge to L as t —> +00 (—cx)) is called the stable (unstable) manifold of the periodic orbit. They are denoted by W and W , respectively. In the case where m = n, the attraction basin of L is Wf. In the saddle case, W is (m + l)-dimensional if m is the number of multipliers inside the unit circle, and is (p + 1)-dimensional where p is the number of multipliers outside of the unit circle, p = n — m — 1. In the three-dimensional Cctse, Wl and are homeomorphic either to two-dimensional cylinders if the multipliers are positive, or to the Mobius bands if the multipliers are negative, as illustrated in Fig. 7.5.1. In the general case, they are either multi-dimensional cylinders diffeomorphic to X S, or multi-dimensional Mobius manifolds. 

Recall that a fixed point 0 x = xq) is called structurally stable if none of its characteristic multipliers, i.e. the roots of the characteristic equation (7.5.2), lies on the unit circle, A topological type (m,p) is assigned to it, where m is the number of roots inside the unit circle and p is that outside of the unit circle. If m = n (m = 0), the fixed point is stable (completely unstable). The fixed point is of saddle type when m 0,n. The set of all points whose trajectories converge to xq when iterated positively (negatively) is called the stable (unstable) manifold of the fixed point and denoted by Wq Wq). In the case where m = n, the attraction basin of O is Wq. If the fixed point is a saddle, the manifolds Wq and Wq are C -smooth embeddings of and MP in respectively. 

Fig. 10.5.6. Each iteration maps the stable (unstable) manifold through a tt/2 angle.

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