Triple collision orbit

Property 1. Triple collision orbits have zero angular momentum. 

As done in Refs. [29,31,32], by using the symbolic dynamics, we can specify the location of the triple colhsion orbits. We skip the detail of this method. The triple collision orbits that experience a triple colhsion in the future will form curves on the Poincare section. Then we call them the triple collision curves. 

The connection between the observation in Fig. 5 and the observation in Figs. 6b and 7b is unknown here. Now we elucidate this connection. Figure 6b for (Z, ) = (1,1) shows one branch of Wtcm c) and the triple collision orbits on the Poincare section. A remarkable point is that the triple collision curves tc and accumulate at 10 points on 0. As shown in Fig. 9a, these points are the points at which if tcm c) and if TCM d) cross the plane x = 0 and which are, of course, just on 0. We denote these points by PTCM,r=o- It is clear that the number of points of Ptcm,t=o is related to the existence of tori in the Poincare section 3>. If the tori exist, its outer most torus has periodic points. These periodic points have the stable and unstable manifolds. Branches of these stable and unstable manifolds run toward precisely if tcm c) and if TCM d) on 0. This situation was observed in Fig. 8. Therefore, the number of the points of Ptcm,t=o is related to the existence of the tori. At the same time, the number of the points of PrcM,r=Q just corresponds to the winding number xF of f tcm c) or ifTcmid) around the body of the TCM as mentioned in the previous subsection. In Figs. 7b and 9b, the case of (Z, = (1,7) is shown. As... 

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