Chaos in a periodic three-particle system under Yukawa interaction

The dynamics of a three-particle motion under singular potentials such as Yukawa and Coulomb provides a Hamiltonian system leading to the billiard system in a high-energy region, which clarifies the interrelation between these two systems. From numerical analyses of phase pictures, trajectories in a real space, winding numbers, and Farey trees, we find several interesting facts: a strong similarity on the topological structure of chaos irrespective of the exchange of the nature of fixed points; a close relation between the symmetry and the winding numbers, i.e., the $C_{3v}$ completely symmetric orbit leads to the coprime rational and the partially symmetric one to the noncoprime one; a classification of the symmetry of orbits on the Farey tree. The results on the symmetry and winding number shed light on the question of the integrability. On the other hand, in a low-energy region, the dynamics is described by the Hénon-Heiles system and hence the billiard motion contains both features of the $C_{3v}$ symmetry and a circle.