Foliated phase space

Furthermore, we investigate the detailed structure of the Poincare surface of section for the case of (Z, ) = (1,1). In this case, there are tori. These tori have the periodic points with period 6 in their outermost part. Here we counted the number of vertices of two triangle, namely 2x3 = 6. These periodic points is associated to one orbit in the whole phase space, which is an antisymmetric orbit in the configuration space. These periodic points have stable and unstable manifolds. In Fig. 8a, we depict the stable manifolds of the these periodic points. In Fig. 8b, we also depict the unstable manifolds by using the symmetry. The stable and unstable manifolds of these periodic points go to It should be noted that the reached points of them on 0 is the accumulation points of and Comparing Figs. 6a and 6b with Fig. 8, it is confirmed that in Figs. 6a and 6b is nearly parallel to the stable manifolds and in Fig. 6b is nearly parallel to the unstable manifolds. Therefore, it is understood that the foliated structure of tc and manifests the foliation of the stable and unstable manifolds—that is, hyperbolic structure. 

It is surely no coincidence then that the symmetries of the lower temperature blue phases ("BPI" and "BPII") are precisely those of the D-surface and the gyroid - Pn3.m and laid respectively. These IPMS, the D-surface and the gyroid, are the most homogeneous "leaves" upon which the foliation of space is built [60]. 

According to Nekhoroshev (1977) and to Morbidelli and Giorgilli (1995), the old and crucial question of stability of a dynamical system turns out to be related to the structure and density of invariant tori which foliate the phase space. For instance the puzzle of the 2/1 gap of the asteroidal belt distribution was explained showing that the corresponding region of the phase space is a weak chaotic one (Nesvorny and Ferraz-Mello 1997). 

If we next increase the energy of the oscillator, the size of the ellipse increases, but there are no qualitative changes. We therefore can visualize the global properties of the harmonic oscillator s phase space quite easily as a series of ellipses snugly nested within one another (as shown in Figure 1). Incidentally, when phase space consists of low-dimensional structures nested in a higherdimensional space (here we have one-dimensional elliptical curves nested in a two-dimensional planar space), the phase space is said to be foliated. 

Next considering the case of reactive motion at the same energy E, we realize that the situation at hand is not very different. The phase space of qi is still elliptical. The phase space of is not elliptical, but it is a simple closed curve, and it still therefore has the same topology as a one-dimensional sphere (every point on a closed curve can be uniquely mapped onto a sphere). Thus, the phase space of reactive motion consists of foliated tori that span both sides of the potential barrier. These reactive tori will be skinny when sliced along the ( 2 Pz) compared to the trapped tori, because they have less energy in the vibrational coordinate and more in the reaction coordinate. In Figure 8 these are labeled Qab j... 

---